(PDF) On Commuting Involution Graphs of Certain Finite Groups

Suppose that G is a finite group and X is a G-conjugacy classes of involutions. The commuting involution graph $${\mathcal {C}}(G,X)$$ C ( G , X ) is the graph whose vertex set is X with $$x, y \in X$$ x , y ∈ X being joined if $$x \ne y$$ x ≠ y and $$xy = yx$$ x y = y x . Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs.

Given group, the commuting graph of, is defined as the graph with vertex set, and two distinct vertices and are connected by an edge, whenever they commute, that is. In this paper we get some parameters of graph theory, as independent number and clique number for groups,. Keywords: independent number, clique number, generalized quaternion group

On Commuting Involution Graphs of Certain Finite Groups A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2017 Ali Aubad School of Mathematics Contents Abstract 5 Copyright Statement 7 Acknowledgements 9 1 Preface 10 2 Background 14 2.1 Group Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Commuting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Commuting Involution Graphs . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Computational Group Theory . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.2 Computer Implementations . . . . . . . . . . . . . . . . . . . . . 28 2.4.3 Randomised Algorithms . . . . . . . . . . . . . . . . . . . . . . . 28 2.4.4 Black-Box Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Commuting Involution Graphs of Double Covers of Sym(n) 30 3.1 Double Covers of Sym(n) . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 x-Graph of Sym(n) and its Double Covers . . . . . . . . . . . . . . . . . 33 3.3 Conjugacy Classes of Involutions of 2. Sn . . . . . . . . . . . . . . . . . . 34 3.4 The Disc Structure of C(2. Sn , X) . . . . . . . . . . . . . . . . . . . . . . 35 3.5 The connectivity of C(2. Sn , X) . . . . . . . . . . . . . . . . . . . . . . . 37 2 3.6 The Diameter of C(2. Sn , X) . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Commuting Involution Graphs of Double Covers of Sporadic Groups and Their Automorphism Groups 66 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 The Mathieu Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Leech Lattice Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5 Monster Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Finite Groups Of Lie-type 88 5.1 Algebraic Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Subgroups of Algebraic Group . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Groups with a BN -pair . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Classification of Simple Algebraic Groups . . . . . . . . . . . . . . . . . 92 5.5 Finite Groups of Lie-type . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6 Commuting Involution Graphs of Exceptional Groups of Lie-type 96 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3 CG (t)-orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4 Disc Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7 Investigation On Commuting Involution Graphs for the Exceptional Groups of Lie-type 2 E6 (2) and E7 (2) 115 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . 116 7.3 The Commuting Involution Graphs of 2 E6 (2) . . . . . . . . . . . . . . . 117 7.4 The Commuting Involution Graphs of E7 (2) . . . . . . . . . . . . . . . 121 7.4.1 Semisimple Classes of E7 (2) . . . . . . . . . . . . . . . . . . . . . 126 7.4.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.4.3 Disks Structure and Orbit Size . . . . . . . . . . . . . . . . . . . 132 7.4.4 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.5 Commuting Involution Graph of 2A . . . . . . . . . . . . . . . . . . . . 134 7.6 Commuting Involution Graph of 2B . . . . . . . . . . . . . . . . . . . . 136 7.7 Commuting Involution Graph of 2C . . . . . . . . . . . . . . . . . . . . 140 8 Conclusions and Future Work A 155 8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.2 Potential for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Magma Implementations 157 B Magma Codes 160 Bibliography 162 Notations and Symbols 169 Abstract The University of Manchester ALI ABD AUBAD Doctor of Philosophy On Commuting Involution Graphs of Certain Finite Groups June 30, 2017 Assume that G is a finite group and X is a subset of G. The commuting graph, denoted by C(G, X), has vertex set X with vertices x, y ∈ X being connected together on the condition of x 6= y and xy = yx. In this thesis, we study these and other related graphs for particular types of finite groups such as the double covers of symmetric groups, double covers of certain finite sporadic simple groups and their autmorphism groups, and several of the exceptional groups of Lie-type. We will pay specific attention to distinguish the discs structure and the diameters for these graphs. 5 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 6 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. 7 iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID= 487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library’s regulations (see http://www.manchester.ac. uk/library/aboutus/regulations) and in The University’s policy on presentation of Theses Acknowledgements In the beginning, I would like to thank my Lord for unending blessings that have been given to me and my family. Many thanks also go to my father Abd Altai and my mother Fadilh Altai, sisters, brothers, my uncle Ahmed Dawood and his wife Sahira Tawfiq, for their constant encouragement and belief in me. Also I express my sincere thanks and appreciation to my wife Arwa Abdullah for her hard work and support and for being generally excellent. My sincere thanks and appreciation to my supervisor Prof. Peter Rowley. Thank you for supporting me, giving me a valuable advice, and your patience during the period of my PhD study. A big Thank you! I am very thankful to The Higher Committee of Education Development in Iraq for their financial support, and everyone at the School of Mathematics at the University of Manchester. It has been an enjoyable, favourable and interesting place to study. Finally, I am very grateful to my group; John Ballantyne, David Ward, Paul Bradley, Tim Crinion, Alex Mcgaw, Peter Neuhaus, Jamie Phillips, Awatef Almotairi and Daniel Vasey. For helping me to adjust to university life and collaborating with their brilliant ideas. Special thanks go to David Ward for his never-ending willingness to help me with everything related to my thesis. Also thanks to the wonderful colleagues in my office; Jacob Cable, Ulla Karhumaki and Zoltan A.Kocsis. They made the office the best place to work. 9 Chapter 1 Preface It is believed that studying the action of a group on a graph is one of the most effective ways of analyzing the structure of the group. From the 1950s era, the method of making the automorphism group of the graph embedded inside the group has provided us with a number of significant results. Currently, many important results have been obtained using these methods. Suppose that G is a group and X is a subset of G; the commuting graph, denoted by C(G, X) has vertex set X and vertices x, y ∈ X are connected on the condition of x 6= y and xy = yx. The commuting graphs were first illustrated by Brauer and Fowler in the seminal paper [21] and they were first for giving evidence that for a given an isomorphism type of an involution centralizer, where there are finitely many non-abelian groups capable of containing it, up to isomorphism. These graphs are vital for work on the Margulis-Platanov conjecture (see [59] as the graphs mentioned in [21] have X = {G \ {1}}). The complement of these graphs is referred to as a non-commuting graph. They also occurred in [54] in which a problem posed by Erdos was solved by B.H. Neumann. Furthermore, as per the assumptions of Maimani, Abdollahi and Akbari, if M is a finite group with a trivial centre and G is a finite simple group such that the non-commuting graphs of G and M are isomorphic then G and M are themselves isomorphic. These sorts of assumptions have been proven in many cases especially for those that satisfy the conjecture of J. Thompson (see [2, 30, 35]). Even while studying the finite simple groups, specifically the non-abelian simple groups, 10 Chapter 1. Preface 11 different sorts of commuting graphs have been established. For instance, Aschbacher and Segev [6] had also used the commuting graph in their computer-free uniqueness proof of the Lyons simple group which has as vertices the 3-central subgroups of order 3. The commuting involution graph is considered a very extraordinary type of commuting graph of G in which the vertex set is a conjugacy class of involutions. At first, the commuting involution graphs were utilized by Fischer in his work when he had to evaluate his work on the 3-transposition groups which was largely unpublished [39, 40]. The vertices were believed to be of the commuting involution graph and were termed to be conjugate involutions in which their product of any two vertices has order at most 3. In addition, this graph led to the construction of sporadic simple groups of Fisher i.e. F i22 , F i23 and F i24 ′ and their detailed structures, and proof the uniqueness are given in [9]. Soon after Aschbacher [8] discovered a condition on a commuting involution graphs of a finite group which ensures the presence of a strongly embedded subgroup was ensured. Rowley, Hart (nēe Perkins), Bates, and Bundy put their efforts into studying the commuting involution graphs and provided the diameters and disc sizes. The entire structure of C(G, X), where X is a conjugacy class of involutions of G, a symmetric group, a finite Coexter group, a sporadic simple group or a projective special linear group has been scrutinized at length by this quartet ([16, 18, 19, 20]). Perkins [57] also studied the commuting involution graphs of Affine Coexter Groups. Moreover, a wider study remarked on the action of the symmetric group on this kind of graph and can be found in [17, 26]. On the contrary, Jafarzadeh and Iranmanesh [47], Mohammadian, Raja, Radjavi and Akbari [4] have also determined the number of commuting graphs through distinct ways. In their works the commuting graphs C(G, X) for a group G has a vertex set of X = G \ Z(G) in which two different elements in X are linked together by an edge if they commute. Nowadays, the commuting graphs are further studied in context of rings (for example [3, 5]). This thesis involves certain discussion regarding the research works of Rowley, Hart, Bates and Bundy in context with the study of commuting involution graphs for symmetric groups [20]. 12 This thesis presents a sequel of sorts to the research of my supervisor Prof. Peter Rowley, in particular his work on the commuting involution graphs of symmetric groups and particular finite simple groups. He provides diameters and scrutinizes the discs structure of the commuting involution graphs for such groups. Here similar work will be investigated, but for certain types of finite groups such as the double covers of symmetric groups, double covers of certain finite sporadic simple groups and their automorphism groups, and several of the exceptional groups of Lie-type. The tools used for this purpose in this project will be theoretical and computational in nature according to the structure of the groups. In Chapter 2, we provide a brief summary of double covers of finite groups, which we will be principally working with in the first four chapters. This chapter will be fundamental in laying the foundations for what will come. A survey of the present research on commuting involution graphs will be given. Finally, general conventions and notation will be explained in this chapter. In Chapter 3, we will the study the commuting involution graphs of double covers of the symmetric group of degree n. The connectivity of this graph will be shown. Additionally we will prove that the diameter of such graphs is at most 5. In Chapter 4, the commuting involution graphs of double covers of certain finite sporadic simple groups and their automorphism groups will be examined. Full information about these graphs will be determined during this chapter. In Chapter 5, we give a brief review of the groups of Lie-type, covering the construction and different widely known properties of such groups. This will be key to the study of the commuting involution graphs of exceptional groups of Lie-type in later chapters. In Chapter 6, we discuss the commuting involution graphs for several of the untwisted groups and the twisted groups. The research involves analyzing the discs structures and calculating the diameters of the graph. In Chapter 7, we examine computationally the suborbit structure of involution conjugacy classes of the exceptional groups of Lie-type 2 E6 (2) and E7 (2) over GF (2). We Chapter 1. Preface 13 apply this input to ascertain the diameter and investigate disc structures of commuting involution graphs. We close this thesis in Chapter 8 by providing a brief summary of our project and present a futuristic vision about related research. Finally, we should mention that in most computational chapters we will use the online Atlas of Group Representations [67] to get a group representation and we refer to it as The Online Atlas. Moreover, the Atlas of Finite Groups Representations [31] plays an important role in this thesis and we refer to it as The Atlas. Chapter 2 Background In this chapter, we will give basic theoretical results and definitions associated to group extensions. Moreover, we supply a literature review of the contemporary research related to commuting graphs and commuting involution graphs that will be used throughout our investigation. 2.1 Group Extensions One of the most frequently asked questions in mathematics is “if we are given a space X, can we extend it to another space Y larger then X?” On the whole, there are two approaches to do that: • By embedding the space X into a space Y which contains at least one an isomorphic copy of X. Equivalently, there is an exact sequence 0→X→Y • The other way is by covering the space X by a space Y which contains an isomorphic copy thereof as a quotient. Equivalently, there is an exact sequence 0→Y →X 14 Chapter 2. Background 15 If X is both embedded in, and covered by, Y , then we call the exact sequence split. Now we present some basic group theoretic results related to the double covers of a group: Lemma 2.1.1 (Dedekind Modular Law). [10] Suppose that A, B and C are subgroups of a group G such that B contains A. Then AC ∩ B = A(B ∩ C). Definition 2.1.2. [44] A complement of a normal subgroup H of a group G is a subgroup K of G such that G = HK and H ∩ K = 1. Definition 2.1.3. [48] A short exact sequence of a groups G, H and K is a sequence i π 1→ − H→ − G− →K where π is an epimorphism and i is a monomorphism such that image i =kerπ. Definition 2.1.4. [48] A group G is called an extension of a group H by a group K if G contains a normal subgroup A such that H ∼ = A and G/A = K. Similarly, there is a short exact sequence i π 1→ − H→ − G− →K Definition 2.1.5. [44] Let G be an extension of a group H by a group K. If H contains a complement in G then, the extension G is called split and denoted by G = H : K. Otherwise, we call G a non-split extension of the group H by the group K and denoted by G = H.K. Definition 2.1.6. [44] Suppose that G and H are groups and π : H → Aut(G) is a group homomorphism. Consider the set S = {(g, a)|g ∈ G, b ∈ H}, and define a binary operation on S as follows (g, a) (k, b) = (gk aπ , ab), for all a, b ∈ H, and for all g, k ∈ G. Then S is a group called a semidirect product of G by H and denoted by G ×π H. The next lemma shows that there is an equivalence between the split extension and semidirect product: 2.1. Group Extensions 16 Lemma 2.1.7. [10] Let G be a group, H a normal subgroup of G and let K be a complement to H in G. Let π : K → Aut(H) be the conjugation map which is defined as follows; for k ∈ K, π(k)(h) = hk for all h ∈ H. Define δ : H ×π K → G by (k, h)δ = kh. Then δ is an isomorphism. The following theorem explains the extension of a normal p-subgroup. Lemma 2.1.8. (Gaschutz Theorem)[10] Let p be a prime, V an abelian normal p − subgroup of a finite group G, and P ∈ Sylp(G). Then G splits over V if and only if P splits over V . Definition 2.1.9. [71] An extension G of a group H by a group K is said to be a central extension if there is a subgroup L of the centre of G such that L ∼ = H. Furthermore, the central extension G of a group H by a group K is called a stem extension if L 6 G′ (where G′ is the derived subgroup of G). Moreover, the stem extension of G is called double, triple, etc., cover if the order of H is 2, 3, etc. Definition 2.1.10. [71] A Schur cover of a finite group G is a group homomorphism π : K → G such that π satisfies the following conditions: 1. kerπ ≤ Z(K) ∩ K ′ , where K ′ is the derived subgroup of K. 2. K has largest size between all homomorphism satisfying (1). The kernel of π is called the Schur multiplier. The group K is often called the Schur cover and π is surjective. Definition 2.1.11. [63] Let G, H be groups we call G and H isoclinic if the following three conditions holds: 1. G/Z(G) ∼ = H/Z(H). 2. The derived subgroup of G is isomorphic to the derived subgroup of H. 3. if a1 Z(G), a2 Z(G) ∈ G/Z(G) with image b1 Z(H), b2 Z(H) under the isomorphism −1 −1 −1 (1) then a1 a2 a−1 1 a2 has image b1 b2 b1 b2 under the isomorphism (2). Chapter 2. Background 17 The relation between the covers and the isoclinic of a finite groups can be seen in the following result: Theorem 2.1.12. [63] All the covering groups for a given finite group are mutually isoclinic. In order to illustrate the above results and definitions we provide the reader the following examples: Examples 2.1.13. • Let G ∼ = H × K, is a direct product of H and K then G is an extension of H by K or K by H. This extension is called the trivial extension. • Suppose that V is a vector space over the field L. Then the general linear group GL(V ) is a central extension of L∗ = L\{0} (the multiplicative group of L) by projective linear group P GL(V ). • The symmetric group Sn and the dihedral group Dn are extensions of An and Zn by Z2 , respectively. • The quaternion group Q8 , Z8 and Z4 × Z2 are extensions of Z2 by Z4 , since there are no subgroups H, K ∈ Z8 or quaternion group Q8 such that H ∼ = Z4 = Z2 , K ∼ with trivial intersection. Therefore, the extensions are non-split in this case, while the extension is split in Z4 × Z2 . 2.2 Commuting Graphs In this section, we will give reviews on the commuting graph C(G, X), including definitions and some essential results related to these kind of graphs. First we will give a summary about graph theory to consolidate our subsequent work follow by the definition of the commuting graph. Let Γ(Ω, E) be a graph with edge set E and vertex set Ω, then Γ(Ω, E) is called an undirected graph if whenever (x, y) ∈ E then (y, x) ∈ E for all x, y ∈ Ω, and with 2.2. Commuting Graphs 18 loops if there is x ∈ Ω such that (x, x) ∈ E. Suppose that d is a distance between any two elements x, y ∈ Ω. We define d(x, y) = j where j is the length of the shortest path between x, y. Hence d is the standard distance metric on Γ(Ω, E). The ith disc of the element x ∈ Ω is defined by ∆i (x) = {y ∈ Ω|d(x, y) = i}. If the length of the 1st disc has the same value for all x ∈ Ω, then the graph Γ(Ω, E) is called a regular graph, and the quantity |∆1 (x)| is call the valency of a regular graph Γ(Ω, E). The graph Γ(Ω, E) is called connected if there is a path between any two vertices in Ω. Finally, if Γ(Ω, E) is a connected graph, the diameter of Γ(Ω, E) is denoted by DiamΓ(Ω, E) = maxx∈Ω {i|∆i (x) 6= φ and ∆i+1 (x) = φ}. Definition 2.2.1. [21] Suppose that G is a group and let X be a nonempty subset of G. The commuting graph of G, denoted by C(G, X), has X as its vertex set with two distinct elements of X joined by an edge whenever xy = yx. The next two theorems, proved by A.Iranmanesh and A.Jafarzadehin in [47], where X taken to be Sn \{1}(n > 2) in 1. and X = An \{1}(n ≥ 5) in 2: Theorem 2.2.2. [47] 1. The commuting graph C(Sn , X) is connected if and only if both n and n − 1 are not prime numbers and in this case Diam(C(Sn , X) 6 5) and the bound is sharp. 2. The commuting graph C(An , X) is connected if and only if neither of n, n − 1 and n − 2 is a prime number and in this case Diam(C(An , X) < 6) and the bound is sharp. The following theorem formulated some results concerning the commuting graph of the dihedral group with a variety of different subsets X: Chapter 2. Background 19 Theorem 2.2.3. [29] Let G = C(D2n , X), where X is a subset of D2n and n > 3. Then the following are true: 1. If X is an abelian subgroup of D2n , then Diam(G) = 1. 2. If X is a non-abelian subgroup of D2n , then Diam(G) = 2. 3. If X = D2n \Z(D2n ), then Diam(G) = ∞ ( the graph is disconnected). Let X be a G-conjugacy class of elements of order 3 in Sn , thus X = tG , where t = (1, 2, 3)(4, 5, 6)(7, 8, 9) . . . (3r − 2, 3r − 1, 3r) of cycle type 1n−3r 3r . Then Athirah Nawawi and Peter Rowley proved the following results: Theorem 2.2.4. [53] 1. If n > 8r, then Diam(C(Sn , X)) = 2. 2. If 6r < n < 8r, then Diam(C(Sn , X)) = 3. 3. If r > 1 and n = 6r, then Diam(C(Sn , X)) 6 4. Now we indicate important results on the commuting graph for finite soluble groups. Theorem 2.2.5. [56] Suppose that G is a finite soluble group with trivial centre and X = G\Z(G). Then 1. C(G, X) is disconnected if and only if G is a Frobenius group or a 2-Frobenius group. 2. If C(G, X) is connected, then C(G, X) has diameter at most 8. Furthermore, there exist soluble groups G with trivial centre such that C(G, X) has diameter 8. Throughout the next results, valuable information about the diameter of the commuting graph C(G, X) is given, where G is a finite non-commutative semigroup and X = G\Z(G): 2.2. Commuting Graphs 20 Theorem 2.2.6. [11] For every n > 2, there is a semigroup G such that the diameter of C(G, X) is n. The complement of the commuting graph is called a non-commuting graph if X = G\Z(G) and it is denoted by Γ(G). Paul Erdos in 1975 [54] was the first author who studied this type of the graph. After that much research appeared about the noncommuting graph (for example [1, 2, 35, 62, 69]). Now we will review some of the results related to non-commuting graph Γ(G), for a variety of groups. Proposition 2.2.7. [52] For every group G, the non-commuting graph Γ(G) is connected. Theorem 2.2.8. [52] Let G be one of the following groups: 1. G = Sn , symmetric group of degree n where n > 3; 2. G = An , alternating group of degree n where n > 4; 3. G is a simple group of Lie type with(t(G) > 2 is number of connected components of Γ(G)). If H is a group such that Γ(G) ∼ = Γ(H). Then |H| = |G|. Proposition 2.2.9. [34] 1. Let G be a non-abelian group, Γ(G) be the non-commuting graph of G and let g ∈ G\Z(G) be an element of order pqr where p, q and r are distinct primes with p < q < r and q ∤ r − 1 and p ∤ r − 1. If H is a group such that Γ(G) ∼ = Γ(H), then |G| = |H|. 2. Let G be a non-abelian group, Γ(G) be the non-commuting graph of G and let g ∈ G\Z(G) be an element of order pn q where p, q are primes where q < p . If H is a group such that Γ(G) ∼ = Γ(H), then |G| = |H|. Theorem 2.2.10. [62] Let G and H be two finite non-abelian nilpotent groups with irregular non-commuting graphs such that Γ(G) ∼ = Γ(H). Then |G| = |H|. Chapter 2. Background 21 The next couple of examples are established to clarify the previous results: Examples 2.2.11. 1. Let G = D16 , the dihedral group of order 16, and X = G \ Z(G). Then C(G, X) is a disconnected commuting graph. Indeed, C(G, X) has 5 connected components four of them have length 2 and one has length 6. 2. Let G = A12 , the alternating group of degree 12, and let X = tG , where t = (12)(34). Then C(G, X) is a connected commuting involution graph, with DiamC(G, X) = 2. Also one can find by using a computational approach that the disc structure of t is as follows 35 268 1 t 1 ∆1 (t) 36 232 ∆2 (t) Where |∆1 (t)| = 268 and |∆2 (t)| = 1216. Also for x ∈ ∆1 (t) and y ∈ ∆2 (t) the number of elements in ∆1 (t) commute with t, x, y respectively equal 268, 35, 232 and the number of elements in ∆2 (t) which commute with x, y is 232, 36 respectively. 2.3 Commuting Involution Graphs In this section we will provide the mathematical definition of the commuting involution graph. Moreover, we furnish an overview about the latest results related to these graphs. A very interesting special case of the commuting graph occurs when the subset X of G is a conjugacy class of involutions. In this case the graph C(G, X) is called the commuting involution graph and this type of graph has a vital role in determining the order of the product of two elements in X. Therefore it is clear that in this graph ∆1 (x) = {y|y ∈ X ∩ CG (x) and x 6= y}. Due to the transitive action of G on X by conjugation, it is clear that C(G, X) is a regular, undirected graph without loops. 2.3. Commuting Involution Graphs 22 This graph was first considered by Peter Rowley and his PhD students and postdoctoral researchers (Chris Bates, David Bundy and Sarah Perkins) in their paper Commuting involution graphs for symmetric groups [19]. In that paper they proved the following results: Theorem 2.3.1. [19] Let a = (1 2)(3 4) . . . (2m − 1 2m) be an involution in Sn and let X = aG . Then C(Sn , X) is disconnected if and only if n = 2m + 1 or n = 4 and m = 1. Theorem 2.3.2. [19] Let a = (1 2)(3 4) . . . (2m − 1 2m) be an involution in Sn and let X = aG . Suppose that C(Sn , X) is connected. Then one of the following holds: i If 2m + 2 = n ∈ {6, 8, 10} then DiamC(Sn , X) = 4, otherwise ii DiamC(Sn , X) 6 3. Corollary 2.3.3. [19] Let a = (1 2)(3 4) . . . (2m−1 2m) be an involution in H ∼ = Alt(n), the alternating group of degree n, and let X = aH . If C(H, X) is connected, then either DiamC(H, X) 6 3 or 2m + 2 = n ∈ {6, 10} and DiamC(H, X) = 4. Definition 2.3.4. [44] A group W is said to be a Coxeter group if the following conditions holds: • W generated by distinct involution wi , 1 6 i 6 m • If wi wi has order kij , then the relations (wi wj )kij = 1 1 6 i, j 6 m. are a complete set of defining relations for W . The next result provides detail about diameter and connectivity of the commuting involution graph C(G, X) where G is a finite Coxeter group. Theorem 2.3.5. [18] Let G be a finite Coxeter group and X a conjugacy class of involutions in G. Chapter 2. Background 23 1. If G is of type Bn or Dn , then C(G, X) is either disconnected or connected of diameter at most 5, with equality in exactly one case. 2. If G is of type E6 , then C(G, X) is connected of diameter at most 5. 3. If G is of type E7 or E8 , then C(G, X) is connected of diameter at most 4. 4. If G is of type F4 , H3 or H4 , then either C(G, X) is disconnected or connected of diameter 2. 5. If G is of type In , then C(G, X) is disconnected. Proof. This is an abridged version of the theorem. See [18] for more information about the proof of this theorem. Soon thereafter, Perkins [57] considered the case of a class of infinite groups and their commuting involution graphs. She proved the following results: Theorem 2.3.6. (Perkins) [57] Let G be an affine Coxeter group of type Ân , and X a conjugacy class of involutions of G. Then C(G, X) is disconnected or is connected of diameter at most 6. Proof. For full details about the steps of the proof we refer to [57]. The next results provide a brief review for the diameter and disc structure of commuting involution graphs of the special linear and projective special linear groups over several fields as mentioned in [20], and then for a 3-dimensional projective special unitary group as given in [36]. The full proofs and details about these results can be found in [36, 37, 38] and [20] respectively, Theorem 2.3.7. (A. Everett)[36, 37] Suppose G ∼ = SU 3(q), the 3-dimensional projective special unitary group over the finite field of q elements such that q = ps , p is an odd prime and s ∈ N, and X a G-conjugacy class of involutions. Let t ∈ X. Then C(G, X) is connected of diameter 3, with disc sizes 2.3. Commuting Involution Graphs 24 |∆1 (t)| = q(q − 1); |∆2 (t)| = q(q − 2)(q 2 − 1); and |∆3 (t)| = (q + 1)(q 2 − 1). Theorem 2.3.8. (A. Everett, P.Rowley )[38] Suppose H ∼ = Sp(4, q), q = pa and p a prime. Let V be the natural (symplectic) GF(q) H-module, and set G ∼ = H/Z(H). So G ∼ = P Sp(4, q) the finite 4-dimensional projective symplectic group. Set V (x) = {v ∈ V |(v, v x ) = 0}. Then we have • If p = 2 and X = {x ∈ G|x2 = 1, dimCV (x) = 3} or X = {x ∈ G|x2 = 1, dimCV (x) = 2, V (x) = V } with t ∈ X. Then C(G, X) is connected of diameter 2, with disc sizes being |∆1 (t)| = q 3 − 2; and |∆2 (t)| = q 3 (q − 1). • If p = 2 and X = {x ∈ G|x2 = 1, dimCV (x) = 2, dimV (x) = 3} with t ∈ X. Then C(G, X) is connected of diameter 4, with disc sizes being |∆1 (t)| = q 2 (2q − 3); |∆2 (t)| = 2q 2 (q − 1)2 ; |∆3 (t)| = 2q 3 (q − 1)2 ; and |∆4 (t)| = q 4 (q − 1)2 . • If p is odd and X is a G-conjugacy class of involution whose elements are the images of an involution in H. Let t ∈ X. Then C(G, X) is connected of diameter 2, with disc sizes being |∆1 (t)| = 21 q(q 2 − 1); and |∆2 (t)| = 12 (q 4 − q 3 + q 2 + q − 2). Chapter 2. Background 25 • If p is odd and X is a G-conjugacy class of involution whose elements are the images of an elements of H of order 4 which square to the non-trivial element of Z(H). Let t ∈ X. – If q ≡ 3 (mod 4). Then C(G, X) is connected of diameter 3. Furthermore, |∆1 (t)| = 12 q(q 2 + 2q − 1); |∆2 (t)| = 1 16 (q + 1)(3q 5 − 2q 4 + 8q 3 − 30q 2 + 13q − 8); and |∆3 (t)| = 1 16 (q − 1)(5q 5 − 4q 4 − 2q 3 + 4q 2 + 5q + 5). – If q ≡ 1 (mod 4). Then C(G, X) is connected of diameter 3. Furthermore, |∆1 (t)| = 12 q(q 2 + 1); |∆2 (t)| = 1 16 (q − 1)(3q 5 − 6q 4 + 32q 3 − 10q 2 − 27q − 8); and |∆3 (t)| = 1 16 (q − 1)(5q 5 + 22q 4 − 8q 3 + 34q 2 + 51q + 24). Theorem 2.3.9. (Bates, Bundy, Perkins, Rowley) [20] Suppose G ∼ = L2 (q), the 2-dimensional projective special linear group over the finite field of q elements, and X a G-conjugacy class of involutions with t ∈ X. (i) If q is even, then C(G, X) consists of q + 1 cliques each with q − 1 vertices. (ii) If q ≡ 3 (mod 4), with q > 3, then C(G, X) is connected and Diam C(G, X)= 3. Furthermore, |∆1 (t)| = 12 (q + 1); |∆2 (t)| = 41 (q + 1)(q − 3); and |∆3 (t)| = 41 (q + 1)(q − 3). (iii) If q ≡ 1 (mod 4), with q > 13, then C(G, X) is connected and Diam C(G, X) = 3. Furthermore, |∆1 (t)| = 12 (q + 1); 2.3. Commuting Involution Graphs 26 |∆2 (t)| = 14 (q − 1)(q − 5); and |∆3 (t)| = 41 (q − 1)(q + 7). Theorem 2.3.10. (Bates, Bundy, Perkins, Rowley) [20] Suppose G ∼ = SL3 (q), and X is the G-conjugacy class of involutions. Let t ∈ X. Then C(G, X) is connected with DiamC(G, X) = 3 and the following hold. (i) If q is even, then |∆1 (t)| = 2q 2 − q − 2; |∆2 (t)| = 2q 2 (q − 1); and |∆3 (t)| = q 3 (q − 1). (ii) If q is odd, then |∆1 (t)| = q(q + 1); |∆2 (t)| = (q 2 − 1)(q 2 + 2); and |∆3 (t)| = (q + 1)(q − 1)2 . Theorem 2.3.11. (Bates, Bundy, Perkins, Rowley) [20] Let K be a (possibly infinite) field of characteristic 2, and suppose that G ∼ = SLn (K) and X a G-conjugacy class of involutions containing t. Also let V denote the natural n-dimensional KGmodule, and set k = dimK [V, t], where [V, t] = {v ∈ V |v t = v}. (i) If n > 4k, then DiamC(G, X) = 2. (ii) If 3k 6 n < 4k, then DiamC(G, X) 6 3. (iii) If 2k < n < 3k, or k is even and n = 2k, then Diam C(G, X) 6 5. (iv) If n = 2k and k is odd, then Diam C(G, X) 6 6. Chapter 2. Background 27 The structure of the commuting involution graph of the sporadic simple groups has taken a large space in recent studies. The following results are related to these kinds of graphs. For a deeper understanding to the notation of the conjugacy classes of involutions in these results we refer the reader to the Atlas[31]. Theorem 2.3.12. (Bates, Bundy, Hart, Rowley; Rowley, Taylor; Rowley)[16] [60] [64] Let K be a sporadic simple group and K 6 G 6 Aut(K). Let X be a conjugacy class of involutions in G. (i) For (K, X) not equal to (J4, 2B),(F i′24 , 2B),(F i′24 , 2D),(B, 2C),(B, 2D) or (M, 2B), the diameter of C(G, X) is at most 4, with equality in precisely four cases. (ii) For (K, X) equal to (J4, 2B), (F i′24 , 2B), (F i′24 , 2D), the diameter of C(G, X) is 3. (iii) For (K, X) equal to (M, 2B) the diameter of C(G, X) is 3. Proof. Bates, Bundy, Hart and Rowley [16] provided the proof of part (i). Rowley and Taylor [64] gave the proof of part (ii). Finally, an unpublished manuscript of Rowley [60] established part(iii). 2.4 Computational Group Theory This section will clarify the general concepts concerning the computational approach that will be utilized to study commuting involution graphs and relevant group theory. 2.4.1 Algorithms The “pseudocode” format used in [45] will be applied to display many of our algorithms. Therefore, we consider standard coding constructs for example, statements like “if ...then” or loops such as “for”. However, the statements will be established in ordinary mathematical language instead of complicated programming language. Also, for a variable y taking a value x we use the notation x 7−→ y. The application of the algorithms can be found in the fourth chapter onwards. 2.4. Computational Group Theory 2.4.2 28 Computer Implementations In computational group theory, the computer algebra systems Magma [23] and GAP [41] are considered to be a most commonly utilized. In most steps of our algorithms, we mainly use Magma, whilst GAP is used for the implementation associated with the character table of the group. This is because GAP provides a comprehensive library of character tables produced from Atlas [31] and is computationally efficient. Moreover, the electronic files associated to this thesis are all set up in Magma format. Appendices A and B have information about Magma functions. 2.4.3 Randomised Algorithms The search for elements possessing certain properties may be impossible in some cases, especially if the search process occurs within groups having a large size. Thus the best available solutions will be found by finding random elements inside such groups and checking if they have the qualities required. This method is known as a “Randomised Algorithm” and the main role for this kind of algorithm will be looking for the orbits that form the foundation on which to build the commuting involution graphs. Implementation of the procedures associated with randomised algorithms will be via Magma packages. 2.4.4 Black-Box Groups The concept of the black-box group was first introduced in 1984 by Babai and Szemerdi see[12], with the purpose of checking basic group theory properties such as commutativity or finding random elements of the group. The elements of the black-box group are designated (not always uniquely) by bit-strings of uniform length N , also there exist black-boxes (oracle) used to perform the group operations. Let g, h be strings in G. Then the black-boxes calculate gh, g −1 and checks whether h = g, each operative performed during a specified period of time. As an example for a black-box group we explain the Bray method established in [22] which aim to calculate the centralizer of an involution. This method is based on the following result proved in [22]. Chapter 2. Background 29 Lemma 2.4.1. [22] Let G be a group and t ∈ G be an involution. For any g ∈ G. Let −1 m be the order of ttg . Then if m is even, (ttg )m/2 , (ttg )m/2 ∈ CG (t) whereas if m is odd, g(ttg )m−1/2 ∈ CG (t). Proof. See [22, section 2.2]. According to the above lemma we build Algorithm 1. Algorithm 1 . Input: G a black-box group, t an involution in G; i: g −→ Random(G), set n =Order(ttg ). ii: if n is even then −1 c1 = (ttg )n/2 and c2 = (ttg )m/2 else; iii: c1 = g(ttg )m−1/2 and c2 = t. Output: c1 , c2 ∈ CG (t). The implementation of Algorithm 1 involves choosing a random element g ∈ G, computing inverses and products when constructing the element ttg and the centralizing elements, which involves computing the order of ttg . Thus Algorithm 1 is black-box. Chapter 3 Commuting Involution Graphs of Double Covers of Sym(n) In this chapter we investigate the commuting involution graphs of double covers of symmetric groups by studying the structure of the graph. We also analyze the connectivity and diameter of the graph. I. Schur [61], proved that for n > 4, the symmetric group Sn has two proper double covers namely Sn+ and Sn− and they are isomorphic only when n = 6. Furthermore, the Schur multiplier of these covers is a cycle group of order 2. We will set t = −1 to be the generator of the centre for this 2-group ( Schur multiplier) so that t is an involution in 2. Sn and 2. Sn / < t >∼ = Sn . In order to cope with C(2. Sn , X) where X is an involution class in 2. Sn we need to have enough information about the product of involutions of 2. Sn . The next section will deal with this for both covers of Sn . 3.1 Double Covers of Sym(n) Definition 3.1.1. (Sn− )[24] It is the double cover of Sn which lifts a transposition of Sn to an element of order 4 in Sn− . This cover of Sn has generators t, r1 , r2 , . . . , rn−1 and relations: t2 = 1; 30 Chapter 3. Commuting Involution Graph of 2. Sn 31 tri = ri t, ri2 = t for i = 1, 2, . . . , n − 1; (rj rj+1 )3 = t for j = 1, 2 . . . , n − 2; and rk rh = t rh rk for |h − k| > 1 and h, k = 1, 2, . . . , n − 1. Definition 3.1.2. (Sn+ )[24] It is the double cover of Sn which lifts a transposition of Sn to an element of order 2 in Sn+ . This cover of Sn has generators t, r1 , r2 , . . . , rn−1 and relations: t2 = 1; tri = ri t, ri2 = 1 for i = 1, 2, . . . , n − 1; (rj rj+1 )3 = 1 for j = 1, 2, . . . , n − 2; and rk rh = t rh rk for |h − k| > 1 and h, k = 1, 2, . . . , n − 1. To deal with a calculations of the Schur double covers we will use the method of Conway and others at Cambridge. This first appears in the Atlas [31]. This method is summarized in papers by David B. Wales[68] and J. Brinkman [24]. In this the elements of 2. Sn are products of the form ±[σi ], where σi are disjoint cycles in Sn . These elements are the two lifts of [σi ] in 2. Sn . Consider the complex algebra of the Clifford Algebra C(Ω) as mentioned in [24] and [68] (a Clifford Algebra is a unital associative algebra generated by a vector space with a quadratic form) where Ω = {1, 2, · · · , n} ∪ δ. Which has the following generators A1 , A2 . . . , Aδ such that A2i = 1 in Sn+ ( A2i = −1 in Sn− ) and Ai Aj = −Aj Ai for i 6= j. The subgroups of these complex algebra generated by √ √ √ (A1 − A2 )/ 2, (A2 − A3 )/ 2, . . . , (An − Aδ )/ 2 √ is isomorphic to 2. Sn . By denoting ai with (Ai − Aδ )/ 2, such that i′ s are distinct in Ω \ {δ}. We obtain immediately the following relations (ai )2 = 1 in Sn+ ((ai )2 = −1 in Sn− ) and a1 a2 · · · am a1 = (−1)m+1 a2 a3 · · · am a1 a2 . For more information regarding Clifford Algebra and their representation we refer the interested reader to [42] and [71]. The idea of this method relies mainly on the following definition: Definition 3.1.3. [68] [24] For the elements a1 , a2 , . . . , am we define 3.1. Double Covers of Sym(n) 32 [a1 a2 . . . am ] = a1 a2 . . . am a1 We call ±[a1 a2 · · · ak ] signed cycles in 2. Sn . Each is a lift of the cycle (a1 a2 · · · ak ) in Sn . Similarly for any permutation k in Sn , we have the signed permutations ±kn in 2. Sn . In fact each ai corresponds to an element of a subgroup of a Clifford algebra which is isomorphic to 2. Sn . The following rules are sufficient to enable the calculation of products of disjoint signed cycles in 2. Sn (these appear as 2.3 and 2.4 in [68] and 3.1 in [24]). 1. [ai ] = t in Sn− and [ai ] = 1 in Sn+ . 2. [a1 a2 · · · am ] = (t)m+1 [a2 a3 · · · am a1 ]. 3. [a1 a2 · · · am−1 ]am = (t)m am [a1 a2 · · · am−1 ]. To be familiar with the above relations we form the following basic example in 2. Sn− : Example 3.1.4. [12][12] = 121121 = 12[1]21 = t1221 = t1[2]1 = t2 [1] = t. The next lemma will give way for explicit computation inside 2. Sn . Lemma 3.1.5. [68] The following relations hold in 2. Sn . 1. [i j][k l] = −[i j] where i, j, k, l are distinct. 2. [i j][j k] = −[i k] where i, j, k are distinct. 3. [i j][i j] = −[j i] = [i j] where i, j are distinct. 4. [i j]k = (sgn k)[u v] where k is a permutation in Sn which maps i to u and j to v and sgn k is the sign of k considered as a permutation in Sn . 5. [i j][i k] = [i j k] = (−[i k j])2 where i, j, k are distinct. Proof. The relations hold in case of the cover Sn− see [68] and for the cover Sn+ may be proved in a similar way. Chapter 3. Commuting Involution Graph of 2. Sn 33 For a transposition (ri rj ) ∈ Sn , we call ±[ri rj ] a signed transposition in 2. Sn . The following proposition shows when the product of disjoint signed transposition for both covers of 2. Sn has order 2. Proposition 3.1.6. [25]. let X be the product of m disjoint signed transposition in 2. Sn . Then 1. if 2. Sn ∼ = Sn− and is m congruent to 0 or 3 mod 4, then X has order 2, otherwise X has order 4. 2. if 2. Sn ∼ = Sn+ and is m congruent to 0 or 1 mod 4, then X has order 2, otherwise X has order 4. 3.2 x-Graph of Sym(n) and its Double Covers . Suppose that G = 2. Sn and a is an involution in G. Set X = a(2 Sn ) , the conjugacy class of involution for a. We may assume without loss of generality that a = [1 2][3 4] · · · [2m − 1 2m][2m + 1][2m + 2] . . . [n], where m is congruent to 0 or 3 mod 4 in Sn− and congruent to 0 or 1 mod 4 in Sn+ (see Proposition 3.1.6). For x ∈ G we define f ix(x) to be the set of fixed point of x on the set Ω = {1, 2, . . . n} . The support of x, which denoted by supp(x) is defined to be the set Ω \ f ix(x). One can see that |f ix(a)| = n − 2m, thus |supp(a)| = 2m. Definition 3.2.1. (x-Graph in Sn ) [19] Let Sn be the symmetric group of degree n, acting by usual manner on the set Ω = {1, 2, · · · , n}. Let b = (1 2)(3 4) · · · (2m − 1 2m)(2m + 1)(2m + 2) . . . (n) be an involution in Sn and Y = bSn . Assume that V = {{1, 2}, {3, 4}, · · · , {2m−1, 2m}, {2m+ 1}, {2m + 2}, . . . , {n}} hence V is the set of hbi-orbit of Ω, and suppose that x ∈ Y . We define the x-graph, denoted by Gx , to be the graph which has V as its vertex set and two subset v1 , v2 in V are connected by an edge if there is α1 ∈ v1 and α2 ∈ v2 with α1 6= α2 , and α1 x = α2 . We call the vertices generated by a transposition in Sn black 3.3. Conjugacy Classes of Involutions of 2. Sn 34 vertices and denote then by (•). Similarly we call the vertices generated by element which lie in f ix(b) white vertices and denote then by (◦). An example for an x-graph let n = 8, m = 3 , b = (12)(34)(56) and x = (13)(24)(76). Then Gx is . In general we denote the x-graph Gx by Gxb , that is the graph that has the orbits of Ω under hbi as a vertex set and the orbits of Ω under hxi as defining its edges. Now ′ let x ∈ X. We may assume that Gxa = Gxa′ , where x′ , a′ are the image of x, a by the covering map respectively. Therefore, it is obvious that x and tx have the same xgraph in 2. Sn . An example for this case, let n = 10, m = 4, a = [12][34][56][78][9][10]and x = [14][23][56][910], then the x-graph has the following connected components . We should note that if Gx has a connected component of the form . Then Gx will produce a connected component of the form corresponding to the component , . The next lemma determines the possible connected components of Gx . Lemma 3.2.2. [19] Let x ∈ X. Then the x-graph has one of the following connected components: 1- , 2- , 3- 3.3 , , , , (all black vertices); (black and white vertices); (all white vertices). Conjugacy Classes of Involutions of 2. Sn The next theorem is essential to determine the size of conjugacy classes in 2. Sn . Theorem 3.3.1. [46] Let H be a group with a central subgroup Z = hzi of order 2 and θ : H −→ H/Z be the natural homomorphism. For any conjugacy class C̄ of H̄ = H/Z, the inverse image θ−1 (C̄) is either a conjugacy class in H or a union of two classes in H. The latter case arises precisely when there is no element of θ−1 (C̄) conjugate to z times itself. Chapter 3. Commuting Involution Graph of 2. Sn 35 The following proposition pins down the size of conjugacy classes of involution in 2. Sn . . Proposition 3.3.2. For n > 4. Let a = [1 2][3 4][5 6] · · · [2m − 1 2m] and X = a2 Sn a conjugacy class of involution in 2. Sn . Let ā be the image of a under the covering map in Sn . If s is the size of the class āSn then 2s is the size of X. Proof. By Proposition 3.1.6 we have m congruent to either 0, 3 or 1 modulo 4 and n = 2m + r where r = |f ix{a}|. Using the above theorem it is enough to prove that ta and a are conjugate. Therefore, if m is congruent to 0 mod 4 then set c = [12] so ac = [12][12][34] . . . [2m − 1 2m]([12])−1 , but m is even we get ac = t[12][12]([12])−1 [34] . . . [2m − 1 2m], thus ac = ta. If m = 1 set c = [3 4] then part 1 of Lemma 3.1.5 shows that ac = ta. The remaining cases are m congruent to 1 or 3 mod 4 and m 6= 2, in this case put c = [13][24], then ac = [1 3][2 4][12][34] . . . [2m − 1 2m]([1 3][2 4])−1 , this lead to ac = [1 3][2 4][12][34]([2 4])−1 [1 3]−1 . . . [2m − 1 2m], so that ac = t[1 3][2 4]([2 4])−1 [1 3]−1 [12][34] . . . [2m − 1 2m]. Therefore, ac = ta . 3.4 The Disc Structure of C(2. Sn , X) The structure of the x-graph of the element x ∈ X plays a crucial role in determining whether x commutes with a or not. We know that, if x ∈ ∆1 (a) ∪ {a}, then this implies x̄ ∈ ∆1 (ā) ∪ {ā} in Sn , where x̄, ā are the image of x, a under the covering map respectively. Therefore, information about the discs structure for C(Sn , X̄) will be of great importance when studying discs structure for C(2.Sn , X), where X̄ is the image of the class X under the cover map. The next lemma provides us enough detail about the discs structure for C(Sn , X̄). Lemma 3.4.1. [19] Let x̄ ∈ X̄. Then x̄ ∈ ∆1 (ā) ∪ {ā} if and only if each connected component of Gx̄ is one of , , , or . 3.4. The Disc Structure of C(2. Sn , X) 36 For x ∈ X the next lemma shows what are the necessary and sufficient conditions on Gx to know that x commutes with a. Lemma 3.4.2. Let x ∈ X. Then x ∈ ∆1 (a) ∪ {a} if and only if Gx has the following connected components: (i) even number, even number, even number + ; (ii) odd number, odd number, odd number + ; (iii) even number + (iv) even number, (v) . ; even number + ; or Proof. By Lemma 3.4.1 we only need to show that the necessary part of the argument. In order to prove (iii) we may assume that k is the number of cycles of length 2 in Gx ( ) . So without loss of generality , we may let a = [1 2][3 4] · · · [2m−1 2m] and x = [1 3][2 4] · · · [4k −3 4k −1][4k −2 4k][4k +1 4k +2][4k +3 4k +4] · · · [2m−1 2m]. Since [[a1 a2 ][a3 a4 ], [a1 ai ][aj ak ]] = t, we have [a, x] = tk where {i, j, k} = {2, 3, 4} and a1 , a2 , a3 , a4 ∈ supp(x). This proves (iii). To prove (iv) we may assume that l is the number of two connected white vertices in Gx ( ). So without loss of generality we let x = [2m + 1 2m + 2][2m + 3 2m + 4] · · · [2(l + m) − 1 2(l + m)][2l + 1 2l + 2][2l + 3 2l + 4] · · · [2m − 1 2m][1][2] . . . [2l][2(l + m) + 1] . . . [n]. Also since [[a1 a2 ], [a3 a4 ]] = t for different ai s. Therefore, [a, x] = tl and this proves (iv). Furthermore, (i),(ii) follow for the proof of (iv) and (iii). Also (v) occurs when a = x. The next example show how Lemma 3.4.2 works Example 3.4.3. Let n = 12, and m = 4, then a = [12][34][56][78]. Suppose that x = [13][24][67][58], y = [12][34][67][910] and w = [13][24][910][1112]. Then Gx , Gy , and Gw , have the following connected components, respectively: ; ; and . Chapter 3. Commuting Involution Graph of 2. Sn 37 Then by Lemma 3.4.2 we have that a commutes with x, whereas a does not commute with w or y. 3.5 The connectivity of C(2. Sn , X) The next lemma gives a way to prove the connectivity of the commuting involution graphs of finite groups. We will denoted by CC the connected component containing t inside C(G, X). Also we will denote by StG (S) the stabilizer in a finite group G of a finite set S. Lemma 3.5.1. Let G be a finite group and X be a conjugacy class of involutions G containing t. Then the graph C(G, X) is connected if and only if G = StG (CC ). Proof. Set S = StG (CC ). If C(G, X) is connected then we must have CC = X. For any g ∈ G and x ∈ CC the element xg ∈ CC so that G = S. To prove the other side of the argument assume that G = S. If t ∈ CC , then as CC S = CC , we have tS ⊆ CC . Here X = CC and the connectivity of the graphs is satisfied. Theorem 3.5.2. Let X be a conjugacy class of involutions in 2. Sn . Then C(2. Sn , X) is disconnected if and only if either n = 2m + 1, n = 6 and m = 3 or n > 4 and m = 1. Proof. If n = 4 and m = 1 or n = 2m + 1, then C(2. Sn , X) is disconnected by Theorem 2.3.1. Thus we may assume that n = 6 and m = 3. In this case Lemma 3.4.2, shows that there is no path between [1 2][3 4][5 6] and [1 3][2 4][5 6]. The same reason shows that if n > 4 and m = 1, there is no path between [1 2] and [1 3] . Therefore, we need to prove for n > 8 that C(2. Sn , X) is a connected commuting involution graph. To prove this, since 2. Sn is generated by elements of the form [α β] such that α, β ∈ Ω. Hence, by Lemma 3.5.1 it is satisfactory to prove a and ar are connected for all element r of the form [α β]. Let r = [α β] where α, β ∈ Ω. Now if α, β ∈ f ix(a) or αa = β then it is understandable that either a = ar or a = tar . In such case clearly there is a path between a and ar . Now suppose that α, β ∈ supp(a) and αa = β1 and β a = β2 . Let a = [1 2][3 4][5 6] · · · [α β1 ][β2 β] · · · [2m − 1 2m] so that ar = ±[1 2][3 4][5 6] · · · [β β1 ][β2 α] · · · [2m − 1 2m]. Since n > 8, we may 3.6. The Diameter of C(2. Sn , X) 38 set w = [1 3][2 4][5 6] · · · [α β][β1 β2 ] · · · [2m − 1 2m] and then Lemma 3.4.2 gives d(a, w) = d(w, ar ) = 1. Finally, assume without loss of generality that α ∈ f ix(a) and β ∈ / f ix(a) so let β a = β1 . Hence without loss of generality let a = [1 2][3 4][5 6] · · · [β1 β] · · · [2m − 1 2m] so that ar = ±[1 2][3 4][5 6] · · · [β1 α] · · · [2m − 1 2m]. If n > 2m + 3 set w = [1 3][2 4][5 6] · · · [c d] · · · [2m − 1 2m] where c, d ∈ f ix(a) \ {α}. Also β1 , α ∈ f ix(w). Thus Lemma 3.4.2 proves that d(a, w) = d(w, ar ) = 1 in this case. On the other hand, if n = 2m + 2, then let w1 = [1 3][2 4][c α] · · · [β1 β] · · · [2m − 1 2m] and w2 = [1 4][2 3][β1 α] · · · [c β] · · · [2m − 1 2m] such that c ∈ f ix(a) and c 6= α. Now by using Lemma 3.4.2 we get d(a, w1 ) = d(w1 , w2 ) = d(w2 , ar ) = 1 ( note that f ix(w1 ) = {5, 6}). This completes the proof of the theorem . 3.6 The Diameter of C(2. Sn , X) In this section we consider the diameters of the commuting involution graphs C(2. Sn , X). The study includes giving specific value to the restrictions of the diameters of the graph. In fact we going to prove that the diameters of the graph is almost 4. Note that for a group G and x, y ∈ G we denoted by [x, y] = x−1 y −1 xy the commutator of x and y in G. Proposition 3.6.1. Let X be a conjugacy class of involution in a double cover of Sn , such that n > 8 and let x ∈ X. If there is no α ∈ f ix(a) such that αx ∈ / f ix(a), then d(a, x) 6 3. Proof. The aim is to find b, b′ ∈ X such that d(a, b) = d(b, b′ ) = d(b′ , x) = 1. Then we have d(a, x) 6 3. Suppose that the connected components of Gx are C1 , C2 , C3 , . . . , Cu . As any connected component Ci of the x-graph Gx is generated by a part from a and a part from x , it is possible to put ai and xi as a represent of the component parts of Ci in Gx . We should note that there are no edges between white and black vertices in the x-graph Gx . Therefore, according to the Lemma 3.2.2 we have to consider the following cases: Chapter 3. Commuting Involution Graph of 2. Sn 39 First, we assume that Ci is a cycle containing only black vertices, put ai = [1 2][3 4] · · · [2k − 1 2k] , and xi = [2 3][4 5] · · · [1 2k]. In order to find a path between ai and xi , we have to consider the following cases which depend on k: Case 1.1: If k = 2. In this case, we may assume without loss of generality that ai = [1 2][3 4] and xi = [2 3][1 4]. Thus the corresponding part in Gx is given by . We have the following subcases: i- if Gx has the following components , then we may write x as follows: Q1 Q2 Q3 Q4 x = [2 3][1 4] ri=1 [βi βi′ ] ri=1 [λi λi + 1] ri=1 [δi ] ri=1 [αi ] , where βi , βi′ , δi ∈ Q2 Q2 f ix(a) and λi , λi + 1, αi ∈ supp(a) such that ri=1 [λi λi + 1] = ri=1 [2i + 3 2i + 4]. To find path between a and x we note the following two cases: • If r1 is odd so that by Lemma 3.4.2 we get d(a, x) = 1 . • If r1 even (if r1 = 0 since n > 8 take b = [2 4][1 3][λ1 λ2 ][λ1 + 1 λ2 + Q2 Q3 Q4 Qr1 −2 1] ri=3 [λi λi + 1] ri=1 [δi ] ri=1 [αi ]), otherwise set b = [1 2][3 4] i=1 Q2 Q3 Q4 [βi βi′ ][βr1 βr1 −1 ][βr′ 1 βr′ 1 −1 ] ri=1 [λi λi + 1] ri=1 [δi ] ri=1 [αi ], thus by Lemma 3.4.2 we obtain [a, b] = [b, x] = 1. ii- If Gx has a connected component Cj such that Cj is a cycle of length 2, therefore, we may assume that aj = [5 6][7 8] and xj = [6 7][8 5] be the generator of Cj in Gx . Now put aij = ai aj = [1 2][3 4][5 6][7 8] and xij = xi xj = [2 3][1 4][6 7][8 5]. The corresponding part in Gx is , so that Lemma 3.4.2 gives d(aij , xij ) = 1. iii- If Gx has a connected component Cj such that Cj is a cycle of length 3 , , then let aj = [5 6][7 8][9 10] and xj = [6 7][8 9][10 5]. Here put aij = [1 2][3 4][5 6][7 8][9 10] and xij = [2 3][1 4][6 7][8 9][10 5]. a b In this case set bij = [1 3][2 4][5 6][7 10][8 9]. Thus Gbijij ,Gxijij are given by d(bij , xij ) = 1. and by Lemma 3.4.2, we obtain d(aij , bij ) = 3.6. The Diameter of C(2. Sn , X) 40 iv- If Gx has a connected component Cj such that Cj is a chain of length one consisting of only black vertices , then let aj = [5 6][7 8] and xj = [6 7][5][8]. In this case Gx has a connected component Cl such that al = [β1 ][β2 ] and xl = [β1 β2 ] where β1 , β2 ∈ f ix(a), so that if aijl = [1 2][3 4][5 6][7 8][β1 ][β2 ] and xijl = [2 3][1 4][6 7][β1 β2 ]. taking bijl = a b ijl [1 3][2 4][5 8][6 7][β1 ][β2 ][5][8], we have that Gbijl ,Gxijl ijl are given respectively by and . Hence aijl and bijl commute, as do bijl and xijl . Thus d(aijl , bijl ) = d(bijl , xijl ) = 1. v- If Gx has a connected component Cj such that Cj is a cycle, then, let aj = [5 6][7 8] · · · [2r + 3 2r + 4] and xj = [6 7][8 9] · · · [2r + 2 2r + 3][5 2r + 4] where r > 3. Now put aij = [1 2][3 4][5 6][7 8] · · · [2r + 3 2r + 4] and a xij = [2 3][1 4][6 7][8 9] · · · [2r + 2 2r + 3][5 2r + 4]. It follows that Gxijij is given by . Therefore, our aim is to find bij a b b b and b′ij (if required) corresponding to aij , xij such that Gbijij , Gxijij , Gb ij′ , Gxijij ′ ij are given by, and the total number of is even. Thus Lemma 3.4.2 gives a path between aij , bij , b′ij and xij . However, in order to do that we need to take into account the following cases: • If r is even such that r = 2s and s is even, set bij = [2 3][1 4][5 6][7 2r + 4][8 2r +3] · · · [r +5 r +6]. In this case the number of a b in Gbijij , Gxijij is s . • If r is even such that r = 2s and s is odd, put bij = [1 2][3 4][5 6][7 2r + a b 4][8 2r + 3] · · · [r + 5 r + 6]. We see that in Gbijij , Gxijij the total number of respectively equal to s − 1, s + 1 . • If r is odd such that r = 2s+1 and s is odd, then let bij = [1 3][2 4][5 6][7 2r + 4][8 2r + 3] · · · [r + 5 r + 6]. The total number of a b in Gbijij , Gxijij is s + 1 . • If r = 2s + 1 and s is even, then let bij = [1 2][3 4][5 6][7 2r + 4][8 2r + Chapter 3. Commuting Involution Graph of 2. Sn 41 3] · · · [r+5 r+6], and b′ij = [2 3][1 4][5 7][6 2r+4][8 2r+3] · · · [r+5 r+6]. b b ′ b ij ij in Gaij ij , Gxij equal to s while in Gb ′ it equals Then the number of ij 2. vi- If Gx has a connected component Cj such that Cj is a chain, then let aj = [5 6][7 8] · · · [2r + 3 2r + 4] and xj = [6 7][8 9] · · · [2r + 2 2r + 3]. In this case Gx has a connected component Cl such that al = [β1 ][β2 ], and xl = [β1 β2 ] where β1 , β2 ∈ f ix(a) and r > 2. Thus let aijl = [1 2][3 4][5 6][7 8] · · · [2r + 3 2r+4][β1 ][β2 ] and xijl = [2 3][1 4][6 7][8 9] · · · [2r+2 2r+3][β1 β2 ][5][2r+4]. a ijl is given by Then Gxijl . Hence, our target is to find bijl and b′ijl (if required) corresponding to aijl , xijl such a b b b ′ ijl ijl ijl that Gbijl , Gxijl ′ , Gxijl are given by, ijl , Gb ijl and the total number of and , is both even so that by Lemma 3.4.2 we may find a path between the corresponding parts aijl , bijl , b′ijl ,and xijl . To deal with that we to consider the following cases: • If r is even such that r = 2s and s is even, then set bijl = [1 2][3 4][5 2r+ 3][6 2r + 4][7 2r + 2] · · · [r + 4 r + 5], and b′ijl = [2 3][1 4][5 2r + 4][6 2r + 3][7 2r + 2][8 2r + 1] · · · [r + 4 r + 5]. In this case the total number of , a b b ijl in Gbijl respectively s, 0 and in Gb ijl′ 2, 0. Moreover, in Gxijl ijl ′ ijl the total number equals s − 1, 1. • If r is even such that r = 2s and s is odd, then let bijl = [1 3][2 4][5 2r + 4][6 2r + 3] · · · [r + 4 r + 5]. The total number of , b a ijl in Gbijl respectively equals s + 1, 0,and in Gxijl ijl the total number equals s, 1. • If r is odd such r = 2s + 1 and s is odd, then let bijl = [2 3][1 4][5 2r + a ijl 4][6 2r + 3] · · · [r + 4 r + 5], so that in Gbijl the total number of , b equal s + 1, 0, while in Gxijl ijl it equals s, 1. • If r = 2s + 1 and s is even, set bijl = [1 2][3 4][5 2r + 4][6 2r + 3] · · · [r + 4 r + 5]. In this case the total number of b in Gxijl ijl is s + 1, 1. , a ijl in Gbijl is s, 0, and 3.6. The Diameter of C(2. Sn , X) 42 Case 1.2: If k = 3, then we may assume without loss of generality that ai = [1 2][3 4][5 6] and xi = [2 3][4 5][1 6], and Gxaii is given by . We need to consider the following subcases: i- if Gx has the following components , then we may write x as follows: Qr1 i=1 [βi βi′ ] Q r2 Q r3 Q r4 where βi , βi′ , δi Q2 Q2 ∈ f ix(a) and λi , λi + 1, αi ∈ supp(a) such that ri=1 [λi λi + 1] = ri=1 [2i + x = [2 3][4 5][1 6] i=1 [λi λi +1] i=1 [δi ] i=1 [αi ], 5 2i + 6]. To find a path between a and x we note the following two cases: • If r1 is odd, set b = [1 2][4 5][3 6] Qr1 i=1 [βi Q r1 βi′ ] Q r2 i=1 [λi Q r2 λi +1] Q r3 i=1 [δi ] Q r4 Q r3 i=1 [αi ], and let b′ = [1 3][2 6][4 β1 ][5 β1′ ] i=2 [βi βi′ ] i=1 [λi λi + 1] i=1 [δi ] Qr4 ′ ′ i=1 [αi ]. Hence by Lemma 3.4.2, we obtain d(a, b) = d(a, b ) = d(x, b ) = 1. • If r1 is even, (if r1 = 0 then as n > 8 we may set b = [1 λ1 + Q2 Q3 Q4 1][2 λ1 ][3 5][4 6] ri=2 [λi λi +1] ri=1 [δi ] ri=1 [αi ], and b′ = [1 3][2 6][4 λ1 ] Q2 Q3 Q4 [5 λ1 + 1] ri=2 [λi λi + 1] ri=1 [δi ] ri=1 [αi ]), otherwise we put b = Q3 Q4 Q1 Q2 [1 2][3 4][5 6] ri=1 [βi βi′ ] ri=1 [λi λi + 1] ri=1 [δi ] ri=1 [αi ], and b′ = Q2 Q3 Q4 Q1 [βi βi′ ][β1 β2 ][β1′ β2′ ] ri=1 [λi λi + 1] ri=1 [δi ] ri=1 [αi ]. [1 2][4 5][3 6] ri=2 Again we have by Lemma 3.4.2, that d(a, b′ ) = d(b′ , b) = d(x, b′ ) = 1. ii- If Gx has a connected component Cj such that Cj is a cycle of length 2, similar to the above case 1.1(iii) there is path between ai and xi . iii- If Gx has a connected component Cj such that Cj is a cycle of length 3, we may let aj = [7 8][9 10][11 12] and xj = [8 9][10 11][7 12]. Here put aij = [1 2][3 4][5 6][7 8][9 10][11 12] and xij = [2 3][4 5][1 6][8 9][10 11][7 12]. a b In this case put bij = [1 2][4 5][3 6][7 8][10 11][9 12], so that Gbijij ,Gxijij are given by d(aij , bij ) = d(bij , xij ) = 1. . Hence by Lemma 3.4.2, we get Chapter 3. Commuting Involution Graph of 2. Sn 43 iv- If Gx has a connected component Cj such that Cj is a chain of length one , then let aj = [7 8][9 10] and consisting of only black vertices xj = [8 9]. In this case Gx has a connected component Cl such that, al = [β1 ][β2 ] and xl = [β1 β2 ] where β1 , β2 ∈ f ix(a). If we let aijl = [1 2][3 4][5 6][7 8][9 10][β1 ][β2 ], and xijl = [2 3][4 5][1 6][8 9][β1 β2 ][7][10], a b ijl also set bijl = [1 2][4 5][3 6][8 9][7 10][β1 ][β2 ], then Gbijl ,Gxijl ijl are given respectively by and . Therefore, Lemma 3.4.2 gives d(aijl , bijl ) = d(bijl , xijl ) = 1. v- If Gx has a cconnected component Cj such that Cj is a cycle, then, let aj = [7 8][9 10] · · · [2r + 5 2r + 6] and xj = [8 9][10 11] · · · [2r + 4 2r + 5][7 2r + 6] where r > 3. Now put aij = [1 2][3 4][5 6][7 8] · · · [2r + 5 2r + 6] and a xij = [2 3][4 5][1 6][8 9][10 11] · · · [2r + 4 2r + 5][7 2r + 6]. Then Gxijij is given . Therefore, our aim is to find bij and b′ij by a b b ′ b (if required) corresponding to aij , xij such that Gbijij , Gxijij , Gb ij′ , Gxijij are given ij by, and the total number of is even. Thus Lemma 3.4.2 gives a path between aij , bij , b′ij and xij . To do that we need to take into account the following cases: • If r = 2s with s is even, then set bij = [1 2][3 4][5 6][7 2r + 6][8 2r + 5][9 2r + 4]...[r + 5 r + 8][r + 6 r + 7], and b′ij = [1 2][4 5][3 6][7 2r + 6][8 2r + 4][9 2r + 5][10 2r + 3][11 2r + 2] · · · [r + 6 r + 7]. In this case the a b ′ b in Gbijij , Gxijij is equal to s, while it is 2 in Gb ij′ . total number of ij • If r = 2s with s is odd, then set bij = [1 2][4 5][3 6][7 2r + 6][8 2r + 5][9 2r + 4] · · · [r + 5 r + 8][r + 6 r + 7], and b′ij = [1 3][4 5][2 6][7 r + 6][2r + 6 r + 7][8 2r + 5][9 2r + 4] · · · [r + 5 r + 8]. The total number of a b ′ b in Gbijij , Gxijij it is equal to s + 1, and is equal 2 in Gb ij′ . ij • If r = 2s + 1 with s is even, then set bij = [1 2][3 4][5 6][7 2r + 6][8 2r + 5][9 2r + 4] · · · [r + 5 r + 8][r + 6 r + 7], and b′ij = [1 2][4 5][3 6][7 2r + 3.6. The Diameter of C(2. Sn , X) 44 6][8 2r + 5] · · · [r + 4 r + 9][r + 5 r + 6][r + 7 r + 8]. The total number of a b ′ b in Gbijij , Gxijij is equal to s, and it is 2 in Gb ij′ . ij • If r = 2s + 1 with s is odd, then set bij = [1 2][4 5][3 6][7 2r + 6][8 2r + 5][9 2r + 4] · · · [r + 5 r + 8][r + 6 r + 7]. In this case the total number of a b in Gbijij , Gxijij is s + 1. vi- If Gx has connected component Cj such that Cj is a chain, then let aj = [7 8][9 10] · · · [2r + 5 2r + 6] and xj = [8 9][10 11] · · · [2r + 4 2r + 5]. In this case Gx has a connected component Cl such that al = [β1 ][β2 ], β2 ] where β1 , β2 ∈ f ix(a) and r > 2. Thus let aijl = and xl = [β1 [1 2][3 4][5 6][7 8][9 10] · · · [2r + 5 2r + 6][β1 ][β2 ] and xijl = [2 3][4 5][1 6][8 9] a ijl [10 11] · · · [2r + 4 2r + 5][β1 β2 ]. Then Gxijl is given by . Hence, our target is to find bijl a b b b ijl ijl ijl and b′ijl (if required) corresponding to aijl , xijl such that Gbijl , Gxijl ′ , Gxijl ijl , Gb ′ ijl are given by, and the total numand ber of is even. Then by Lemma 3.4.2 we get paths between aijl , bijl , b′ijl and xijl . For that we consider the following cases: • If r = 2s and s is even, then set bijl = [1 2][3 4][5 6][7 2r + 6][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7] and b′ijl = [1 2][3 6][4 5][β1 β2 ][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7]. In this case the total number of a b ′ , b ijl ijl in Gbijl , Gxijl ′ the numbers ijl respectively equals s, 0. Moreover in Gb ijl equal {1, 1}. • If r = 2s and s is odd, then set bijl = [1 2][3 6][4 5][7 2r + 6][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7]. It follows the total number of a , b ijl in Gbijl , Gxijl ijl respectively {s + 1, 0} and {s, 1}. • If r = 2s + 1 with s is even, then take bijl = [1 2][3 6][4 5][7 2r + 6][8 2r + 5] · · · [r + 5 r + 8][β1 β2 ] and b′ijl = [2 3][1 6][4 5][β1 β2 ][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7]. Hence the total number of , in Chapter 3. Commuting Involution Graph of 2. Sn a b 45 ′ b ijl Gbijl , Gb ijl′ , Gxijl ijl respectively equals {s + 1, 1}, {1, 1} and {s, 0}. ijl • If r = 2s+1 with s is odd, let bijl = [1 2][3 6][4 5][7 2r+6][8 2r+5] · · · [r+ 5 r+8][r+6 r+7] and b′ijl = [1 3][2 6][4 5][β1 β2 ][8 2r+5] · · · [r+6 r+7]. Thus the total number of b , a b ′ ijl in Gbijl , Gxijl ijl is {s + 1, 0}, and in Gb ijl′ is {1, 1}. ijl Case 1.3: If k > 3, then Gxaii is given by . Therefore, our aim is to find bi and b′i if required corresponding to ai , xi such that Gbaii , Gxbii , Gbbi′ , Gxbii are given by ′ i and the total number of is even. Thus Lemma 3.4.2 gives paths between ai , bi , b′i and xi . We need to take into account the following cases: i- If k = 2s and s is even, then bi = [1 k + 1][2 k + 2][3 k + 3][4 k + 4] · · · [k 2k]. In this case the total number of in Gbaii , Gxbii is equal to s. ii- If k = 2s and s is odd, then set bi = [1 2][4 2k][3 2k − 1][5 2k − 2] · · · [k − 1 k + 4][k k + 3][k + 1 k + 2] and b′i = [1 2][3 2k][4 2k − 1] · · · [k − 1 k + 4][k k + 1][k + 2 k + 3]. Here the total number of in Gbaii , Gxbii is equal ′ to s − 1, and in Gbbi′ 2. i iii- If k = 2s + 1 and s is odd, then set bi = [1 2][3 4][2k − 1 2k][5 2k − 2][6 2k − 3] · · · [k − 2 k + 5][k − 1 k + 4][k k + 3][k + 1 k + 2] and b′i = [1 2][3 2k][4 2k − 1][5 2k − 2] · · · [k − 2 k + 5][k − 1 k][k + 3 k + 4][k + 1 k + 2]. Then the total number of in Gbaii , Gxbii is equal to s − 1, while in Gbbi′ ′ i 2. iv- If k = 2s + 1 and s is even, then put bi = [1 2][3 2k][4 2k − 1] · · · [k + 1 k + 2], so that the total number of in Gbaii , Gxbii is s. Secondly, if Ci is a chain containing only black vertices, then we may write ai = [1 2][3 4] · · · [2k − 1 2k] , and xi = [2 3][4 5] · · · [2k − 2 2k − 1]. We note that Gx has a connected component Cl such that al = [β1 ][β2 ] and xl = [β1 β2 ], where β1 , β2 ∈ f ix(a). Thus let ail = [1 2][3 4] · · · [2k −1 2k][β1 ][β2 ] and xil = [2 3][4 5] · · · [2k −2 2k −1][β1 β2 ]. 3.6. The Diameter of C(2. Sn , X) 46 It follow that Gxailil is given by , so that in order to find a path between ail and xil , we have to consider the following cases: Case 2.1: If k = 2, we may assume that ail = [1 2][3 4][β1 ][β2 ] and xil = [2 3][β1 β2 ]. In this case Gxailil is , so that to find a path between them, we need to argue the following cases: i- if x = [2 3] Qr1 i=1 [µi µ′i ] Qr2 i=1 [λi λi + 1] Q r3 i=1 [δi ] Qr4 where λi , λi + 1, δi Qr2 i=1 [λi λi +1] = i=1 [2i+3 2i+ Q r2 i=1 [αi ], ∈ supp(a) and αi , µi , µ′i ∈ f ix(a) such that Q3 Q2(r1 +r2 )+2 4] and ri=1 [δi ] = [1][4] i=2r [i] if r1 6= 1. Without loss of generality 2 +5 we may let [δ1 ] = [1], [δ2 ] = [4] and [β1 β2 ] = [µ1 µ′1 ]. Then Gx has the following components . Thus to find path between a and x we need to consider the following sub cases: • if r1 is odd (if r1 = 1 then since n > 8 we let b = [2 3][1 4][λ1 λ2 ][λ1 + Q2 Q4 Q1 1 λ2 +1] ri=2 [λi λi +1] ri=1 [αi ][β1 ][β2 ] ), we set b = [2 3][1 4][δ3 δ4 ] ri=3 Q2 Q3 Q4 [µi µ′i ] ri=1 [λi λi + 1] ri=5 [δi ] ri=1 [αi ][µ1 ][µ′1 ][µ2 ][µ′2 ], where δ3 , δ4 ∈ (supp(a)∩f ix(x))\{1, 4}. Thus Lemma 3.4.2 gives d(a, b) = d(b, x) = 1. Q2 Q1 [λi λi + • if r1 is even then put b = [δ3 δ4 ] ri=3 [µi µ′i ][µ1 µ2 ][µ′1 µ′2 ] ri=1 Q4 Q3 1] ri=5 [δi ][1][4][2][3] ri=1 [αi ]. Again δ3 , δ4 ∈ (supp(a) ∩ f ix(x)) \ {1, 4} , so d(a, b) = d(b, x) = 1 by Lemma 3.4.2. ii- If Gx has a connected component Cj such that Cj is a cycle of length 2, we may assume that aj = [5 6][7 8] and xj = [6 7][8 5]. Put ailj = ai al aj = [1 2][3 4][5 6][7 8][β1 ]β2 ] and xilj = xi xl xj = [2 3][β1 β2 ][6 7][8 5]. Now a b ilj let bilj = [2 3][1 4][5 7][6 8], so that Gbilj ,Gxilj ilj are given respectively by and . Hence by Lemma 3.4.2 one can see that d(ailj , bilj ) = d(bilj , xilj ) = 1. iii- If Gx has a connected component Cj such that Cj is a cycle of length 3 , then let aj = [5 6][7 8][9 10] and xj = [6 7][8 9][10 5]. Here put ailj = [1 2][3 4][5 6][7 8][9 10][β1 ][β2 ] and xilj = [2 3][β1 β2 ][6 7][8 9][10 5]. Chapter 3. Commuting Involution Graph of 2. Sn 47 a b ilj Let bilj = [2 3][1 4][5 6][7 10][8 9], so that Gbilj ,Gxilj ilj are given respectively by and . Therefore, Lemma 3.4.2 demonstrates that d(ailj , bilj ) = d(bilj , xilj ) = 1. iv- If Gx has a connected component Cj such that Cj is a chain , then let aj = [5 6][7 8] and xj = [6 7]. Similar to above we can find a connected component Cq such that aq = [β3 ][β4 ] and xq = [β3 β4 ] where β3 , β4 ∈ f ix(a). Thus if ailjq = [1 2][3 4][5 6][7 8][β1 ][β2 ][β3 ][β4 ] and xiljq = a b iljq [2 3][β1 β2 ][6 7][β3 β4 ], and we let biljq = [2 3][1 4][5 8][6 7], then Gbiljq ,Gxiljq iljq are given respectively by and . Con- sequently using Lemma 3.4.2 we have d(ailjq , biljq ) = d(biljq , xiljq ) = 1. v- If Gx has a connected component Cj such that Cj is a cycle with aj = [5 6][7 8] · · · [2r + 3 2r + 4] and xj = [6 7][8 9] · · · [5 2r + 4] where r > 3, then let ailj = [1 2][3 4][5 6][7 8] · · · [2r + 3 2r + 4] and xilj = a ilj [2 3][β1 β2 ][6 7][8 9] · · · [2r + 2 2r + 3][5 2r + 4]. Here Gxilj is given by . Hence, our target is to find bilj and b′ilj (if a b b ′ b ilj ilj ilj required) corresponding to ailj , xilj such that Gbilj , Gxilj ′ , Gxilj are given ilj , Gb ilj by, and the total number of and is even. Hence Lemma 3.4.2 gives edges between aijl , bijl , b′ijl ,and xijl . Therefore, to deal with above we need the following cases: • If r is even such that r = 2s and s is even, then set bilj = [1 3][2 4][5 6][7 2r+ 4][8 2r+3] · · · [r+5 r+6] and b′ilj = [1 4][2 3][6 7][5 2r+4][8 2r+3] · · · [r+ a 5 r + 6]. In this case the total number of b b ′ ilj , in Gbilj , Gxilj ilj respec- tively equals {s, 0},{s − 1, 1}. Moreover, in Gb ilj′ equals 2, 0. ilj • If r is even such that r = 2s and s is odd, then set bilj = [1 3][2 4][5 r + 5][6 r + 6][7 2r + 4][8 2r + 3] · · · [r + 3 r + 8][r + 4 r + 7] and b′ilj = 3.6. The Diameter of C(2. Sn , X) 48 [1 4][2 3][5 6][7 2r + 4][8 2r + 3] · · · [r + 5 r + 6]. Consequntly the total , number of b a b ′ ilj in Gbilj , Gxilj ilj respectively equals {s + 1, 0},{s, 1}, while in Gb ilj′ is 2, 0. ilj • If r is odd such that r = 2s+1 and s is odd, put bilj = [1 4][2 3][5 6][7 2r+ 4][8 2r + 3] · · · [r + 5 r + 6]. Here the total number of a , in b ilj Gbilj , Gxilj ilj respectively {s + 1, 0},{s, 1}. • If r = 2s + 1 and s is even, then let bilj = [1 4][2 3][5 6][7 8][2r + 3 2r + 4][9 2r + 2] · · · [r + 5 r + 6] and b′ilj = [1 4][2 r + 5][3 r + 6][5 6][7 2r + a b ilj , in Gbilj , Gxilj ilj 4][8 2r + 3] · · · [r + 4 r + 7]. The total number of b ′ respectively {s, 0},{s + 1, 1}, and in Gb ilj′ 2, 0. ilj vi- If Gx has a connected component Cj such that Cj is a chain with aj = [5 6][7 8] · · · [2r + 3 2r + 4] and xj = [6 7][8 9] · · · [2r + 2 2r + 3], then the x-graph Gx has a connected component Cq such that aq = [β3 ][β4 ] and xq = [β3 β4 ] where β3 , β4 ∈ f ix(a) and r > 2. Now let ailjq = [1 2][3 4][5 6][7 8] · · · [2r+3 2r+4][β1 ][β2 ][β3 ][β4 ] and xiljq = [2 3][β1 β2 ][6 7] a iljq is given by [8 9] · · · [2r + 2 2r + 3][β3 β4 ]. The graph Gxiljq . Hence, our target is to find biljq and b′iljq (if required) corresponding to a b b ′ b iljq iljq iljq ailjq , xiljq such that Gbiljq , Gxiljq ′ , Gxiljq are given by, iljq , Gb iljq and the total number of and is even. Hence Lemma 3.4.2 gives paths between ailjq , biljq , b′iljq and xiljq . We need to take into account the following cases: • If r is even such that r = 2s and s is even, then set biljq = [1 2][3 4][5 2r+ 4][6 2r + 3][7 2r + 2] · · · [r + 2 r + 7][r + 3 r + 6][r + 4 r + 5], and b′iljq = [2 3][1 4][5 2r +4] · · · [r +1 r +8][r +2 r +3][r +6 r +7][r +4 r +5]. In this case the total number of , b a b ′ iljq in Gbiljq , Gxiljq iljq respectively equals {s, 0},{s − 2, 2}. Moreover in Gb iljq′ it equals 2, 0. iljq • If r is even such that r = 2s and s is odd, then let biljq = [1 4][2 3][5 2r+ 4][6 2r + 3] · · · [r + 4 r + 5], so that the total number of , in Chapter 3. Commuting Involution Graph of 2. Sn a 49 b iljq Gbiljq , Gxiljq iljq respectively {s + 1, 0},{s − 1, 2}. • If r is odd such r = 2s + 1 and s is odd, then put biljq = [1 4][2 3][5 2r + 4][6 2r + 3][7 2r + 2] · · · [r + 2 r + 7][r + 3 r + 6][r + 4 r + 5] and b′iljq = [β1 β2 ][β3 β4 ][5 2r + 4][6 2r + 3][7 2r + 2] · · · [r + 2 r + 7][r + a b iljq , in Gbiljq , Gxiljq iljq 3 r + 6][r + 4 r + 5]. Then the total number of b ′ respectively {s + 1, 0},{s, 1}, and in Gb iljq′ 0, 2. iljq • If r = 2s + 1 and s is even, let biljq = [1 2][3 4][5 2r + 3][6 2r + 4][7 2r + 2] · · · [r + 4 r + 5] and b′iljq = [1 4][2 3][5 2r + 4][6 2r + 3][7 2r + 2][8 2r + 1] · · · [r + 4 r + 5]. The total number of a b ′ , in b iljq iljq Gbiljq , Gxiljq ′ the total numbers iljq respectively {s, 0},{s, 2}. Also in Gb iljq is 2, 0. Case 2.2: If k > 2, our target is to find bil and b′il (if required) corresponding to ail , xil such that Gbailil , Gxbilil , Gbbil′ , Gxbilil are given by, ′ il and the total number of and is even. Lemma 3.4.2 then gives a path between ail , bil , b′il and xil . However, to cope with that we need to consider the following cases: i- If k = 2s and s is even, then set bil = [1 2k][2 2k − 1][3 2k − 2] · · · [k k + 1]. In this case the total number of , in Gbailil , Gxbilil respectively equals {s, 0},{s − 1, 1}. ii- If k = 2s and s is odd, then let bil = [1 2][2k − 1 2k][3 2k − 2][4 2k − 3] · · · [k − 2 k + 3][k − 1 k + 2][k k + 1] and b′il = [1 2k][2 2k − 1][3 2k − 2] · · · [k − 3 k + 4][k − 2 k − 1][k + 2 k + 3][k k + 1]. Hence the total number of , in Gbailil , Gxbilil respectively {s − 1, 0},{s − 2, 1}. Moreover in Gbbil′ the total ′ il numbers 2, 0. iii- If k = 2s + 1 and s is odd, then if k = 3 let bil = [1 5][2 6][β1 β2 ] and b′il = [1 6][2 5][3 4]. Otherwise set bil = [1 2k − 1][2 2k][3 2k − 2][4 2k − 3][5 2k − 4] · · · [k − 1 k + 2][β1 β2 ] and b′il = [1 2k][2 2k − 1][3 2k − 2][4 2r − 3][5 2k − 4] · · · [k k + 1]. The total number of , in Gbailil , Gxbilil ′ 3.6. The Diameter of C(2. Sn , X) 50 respectively {s, 1}, whilst in Gbbil′ equals 1, 1. il iv- If k = 2s + 1 and s is even, then let bil = [1 2k − 1][2 2k][3 2k − 2] · · · [k − 1 k + 2][k k + 1] and b′il = [1 2k][2 2k − 1][3 2k − 2] · · · [k − 2 k + 3][k − 1 k][k + , in Gbailil , Gxbilil respectively equals ′ 1 k + 2]. Here the total number of {s, 0},{s − 1, 1}. Additionally in Gbbil′ the number of such components is 2, 0. il Finally, If Ci has non-conected black vertices, so that Gxaii is given by following , and ai and xi are the corresponding parts of Ci , we may let ai = Q1 [1 2][3 4] · · · [2k − 1 2k] and xi = ri=1 [βi βi′ ], where βi , βi′ , ∈ f ix(a). To find a path between ai and xi we consider the following cases: Case 3.1: If r1 is even, then by Lemma 3.4.2 we get d(ai , xi ) = 1. Case 3.2: If r1 6= 1 is odd, then set bi = [1 3][2, 4] of , Qr1 −2 i=1 [βi βi′ ]. In this case the total number in Gbaii , Gxbii respectively equals {1, r1 − 2},{0, 2}. Case 3.3: If r1 = 1, we may assume that ai = [1, 2] and xi = [2m + 1 2m + 2]. Here we only need to care about the following subcases: 1: If Gx has a connected component Cj such that Cj is a cycle, let aj = [3 4][5 6] · · · [2r + 1 2r + 2] and xj = [4 5][6 7] · · · [2r 2r + 1][3 2r + 2] where r > 3. Now put aij = [1 2][3 4][5 6][7 8] · · · [2r + 1 2r + 2] and a xij = [2m + 1 2m + 2][4 5][6 7][8 9] · · · [2r 2r + 1][3 2r + 2]. The graph Gxijij is . Hence, our target is to find bij and b′ij if given by a b b b ′ required corresponding to aij , xij such that Gbijij , Gxijij , Gb ij′ , Gxijij are given by, ij and the total number of and is even. Thus Lemma 3.4.2 gives edges between aij , bij , b′ij and xij . To do that we need to take into account the following cases: i- If r is even such that r = 2s and s is even, set bij = [2m + 1 2m + 2][3 4][5 2r + 2][6 2r + 1][7 2r] · · · [r + 3 r + 4]. In this case the total Chapter 3. Commuting Involution Graph of 2. Sn , number of 51 b a in Gbijij , Gxijij respectively {s − 1, 1},{s, 0}. ii- If r is even such that r = 2s and s is odd, put bij = [1 2][3 4][5 2r + 2][6 2r + 1][7 2r] · · · [r + 3 r + 4]. Thus the total number of a , in b Gbijij , Gxijij respectively {s − 1, 0},{s, 1}. iii- If r is odd such that r = 2s + 1 and s is odd, let bij = [2m + 1 2m + 2][3 2r + 2][4 2r + 1][5 2r] · · · [r + 2 r + 3], and bij ′ = [2m + 1 3][2m + 2 2r + 2][4 2r][2r + 1 5][6 2r − 1] · · · [r + 2 r + 3]. Thus the total number of , a b ′ b in Gbijij , Gxijij , Gb ij′ respectively {s, 1},{s + 1, 0},{2, 0} . ij iv- If r = 2s+1 and s is even, then let bij = [1 2][3 4][5 2r+2][6 2r+1] · · · [r+ 3 r + 4] and bij ′ = [2m + 1 2m + 2][3 5][4 2r + 2][6 2r + 1] · · · [r + 3 r + 4]. , The total number of a b ′ b in Gbijij , Gxijij is {s, 0}, whilst in Gb ij′ is ij {1, 1} . 2: If Gx has a connected component Cj such that Cj is a chain, then let aj = [3 4][5 6] · · · [2r + 1 2r + 2] and xj = [4 5][6 7] · · · [2r 2r + 1] where r > 2. The x-graph Gx has a connected component Cq such that aq = [β1 ][β2 ] and xq = [β1 β2 ] where β1 , β2 ∈ f ix(a). Now let aijq = [1 2][3 4][5 6][7 8] · · · [2r + 1 2r + 2][2m + 1][2m + 2][β1 ][β2 ] and xijq = a ijq is given by [2m + 1 2m + 2][β1 β2 ][4 5][6 7][8 9] · · · [2r 2r + 1]. Then Gxijq . Hence, our target is to find bijq and b′ijq (if rea b b b ′ ijq ijq ijq quired) corresponding to aijq , xijq such that Gbijq , Gxijq ′ , Gxijq are given ijq , Gb ijq by, and the total number of and is even. Therefore Lemma 3.4.2 gives paths between aijq , bijq , b′ijq and xijq .Thus to deal with that we need to check the following cases: i- If r is even such that r = 2s and s is even, set bijq = [1 2][3 2r + 2][4 2r + 1][5 2r] · · · [r + 2 r + 3] and bijq ′ = [1 3][2 2r + 2][4 5][2r 2r + 1][6 2r − 1] · · · [r + 2 r + 3]. In this case the total number of , 3.6. The Diameter of C(2. Sn , X) b a ′ 52 b ijq ijq , Gxijq in Gbijq ′ respectively equals {s, 0},{s − 2, 2},{2, 0}. ijq , Gb ijq ii- If r is even such that r = 2s and s is odd, put bijq = [2m + 1 2m + 2][3 2r + 2][4 2r + 1][5 2r] · · · [r + 2 r + 3] and bijq ′ = [2m + 1 r + 3][2m + 2 r + 2][3 2r + 2][[5 2r + 1][4 2r][6 r + 5] · · · [r + 1 r + 4]. Thus the total a b ′ b ijq ijq , Gxijq in Gbijq ′ is {2, 0} . ijq is {s, 1} and in Gb , number of ijq iii- If r is odd such that r = 2s + 1 and s is odd, let bij = [2m + 1 2m + 2][3 2r + 2][4 2r + 1][5 2r] · · · [r + 2 r + 3], so that the total number of , a b ijq in Gbijq , Gxijq ijq is {s, 1}. iv- If r = 2s + 1 and s is even, then let bij = [1 2][3 2r + 2][4 2r + 1][5 2r] · · · [r + 2 r + 3]. Therefore the total number of a , in b ijq , Gxijq Gbijq ijq respectively equals {s, 0},{s, 2}. Now, let x ∈ X and suppose that Λ1 , Λ2 , Λ3 , Λ4 , are the number of the double edges , cycles with three edges and unconnected black vertices , chains with one edges of black vertices , respectively. Then in order to find a path between the components of a and x we need to consider first the following : Case 4.1: If Λ1 + Λ2 + Λ3 + Λ4 equal an even number. To find a path between a and x, such that d(x, a) = 1 we have the following subcases: i- If Λ1 + Λ2 + Λ3 + Λ4 = 2. Then by Case 1.1, Case 1.2, Case 2.1, Case 3.1 and Case 3.2 there is a path between a and x. ii- If Λ1 + Λ2 + Λ3 + Λ4 = r ≥ 4, such that r is even , then r = r1 + 2 where r1 is an even number. Thus by the above we get a path between the components of a and x. Case 4.2: If Λ1 + Λ2 + Λ3 + Λ4 equal an odd number. To find a path between a and x, such that d(x, a) = 1 we have the following subcases: i- If Λ1 + Λ2 + Λ3 + Λ4 = 1, then by above results one can easily find a path between a and x, such that d(x, a) = 1. Chapter 3. Commuting Involution Graph of 2. Sn 53 ii- If Λ1 + Λ2 + Λ3 + Λ4 = 3 and if three of the Λi′ s equal zero. Then without loss of generality we may assume Λ2 = Λ3 = Λ4 = 0, and Λ1 = 3 (if one of the Λi′ s = 3 for i ∈ {2, 3, 4} then similar considerations can apply to show that d(x, a) = 1). Let a1 = [1 2][3 4], a2 = [5 6][7 8],a3 = [9 10][11 12] and x1 = [2 3][1 4], x2 = [6 7][5 8], x3 = [10 11][9 12]. Set a123 = [1 2][3 4][5 6][7 8][9 10][11 12] and x123 = [2 3][1 4][6 7][5 8][10 11][9 12], 123 is given by so that the x-graph Gxa123 . Let b123 = [1 2][3 4][6 8][5 7][10 11][9 12]. In this case the total number of 123 , in Gba123 , Gxb123 equals 2. 123 iii- If two or one of Λi′ s equal zero then by the same argument as in [ii] we can find a path in this case. iv- If Λ1 + Λ2 + Λ3 + Λ4 = r > 3, such that r is odd, then r = r1 + 3, where r1 is an even number and again by the above we have a path between the components of a and x, such that d(a, x) = 1. Finally, let b and b′ be the product of all the bi′ s and bi′ ‘s , as mentioned above. It is not hard to see that |f ix(a)| = f ix(b)|=|f ix(b′ )|. Then one can see immediately that b, b′ ∈ X. Additionally, since the component parts of a, b, b′ and x are connected consequently d(a, b) = d(b, b′ ) = d(b′ , x) = 1. Hence we conclude that d(a, x) 6 3. Corollary 3.6.2. For n = 2m > 8, DiamC(2. Sn , X) 6 3. Theorem 3.6.3. Suppose that n > 2m + 2 and n > 8. We have DiamC(2. Sn , X) 6 4. Proof. For n = 9, 10 and m = 3, we use Magma to calculate then discs structure of C(2. Sn , X). We find DiamC(2. S9 , X) = 4 with ∆1 (t) = 37, ∆2 (t) = 360, ∆3 (t) = 1602 and ∆4 (t) = 520 where |X| = 2520. Also DiamC(2. S10 , X) = 3 with ∆1 (t) = 91, ∆2 (t) = 1872 and ∆3 (t) = 4336 where |X| = 6300. Thus we may assume that n > Q1 Q2 Q3 11. For any x ∈ X, it is possible to write x as follows: x = ri=1 [αi αi′ ] ri=1 [δi δi′ ] ri=1 Q4 Q5 [βi βi′ ] ri=1 [γi ] ri=1 [λi ], such that αi , αi′ , βi , γi ∈ supp(a) while βi′ , δi , δi′ , λi ∈ f ix(a). Also we have r4 + r5 > 2 as n > 2m + 2. For the purpose of finding a path between a 3.6. The Diameter of C(2. Sn , X) 54 and x, we will show there is b ∈ X with no black vertices connected to white vertices of Gba beside commutes with x. Thus by Proposition 3.6.1 yields the existence of a path between a and b and hence between a and x. However, to deal with this argument we will consider the following cases: i- If r3 = 2s + 1 with s odd, then set b=( Qr1 i=1 [αi αi′ ] Qr2 i=1 [δi δi′ ])[β1 β2 ][β3 β4 ] · · · [βr3 −2 βr3 −1 ][β1′ β2′ ][β3′ β4′ ] · · · [βr′ 3 −2 βr′ 3 −1 ][µ1 µ2 ] where µ1 , µ2 ∈ supp(a) ∩ f ix(x). In this case the total number of , in Gxb is {s, 1}. ii- If r3 = 2s + 1 and s even, we have the following cases: • if r1 = r2 , then we consider to possibilities: (a) If r3 = 1, then since n > 11, we let b = [α1 α1′ ][µ1 µ2 ][µ3 µ4 ] where µ1 , µ2 ∈ supp(a) ∩ f ix{x} and µ3 , µ4 ∈ f ix(a) ∩ f ix{x}. Thus the total , number of in Gxb {0, 2}. (b) if r3 6= 1, we set b = [α1 α1′ ][β1 β2 ][β3 β4 ] · · · [βr3 −2 βr3 −1 ][β1′ β2′ ] [β3′ β4′ ] · · · [βr′ 3 −2 βr′ 3 −1 ][µ1 µ2 ][µ3 µ4 ], where µ1 , µ2 , µ3 , µ4 ∈ supp(a) ∩ f ix(x). Then b commutes with x. The total number of , in Gxb is {s, 2}. • If r1 + r2 = 1, then x is not an involution element in this case this because r1 + r2 + r3 = 2s + 2 = m and m not congruent to either 0,1 or 3 mod 4 in this case contradiction to Proposition 3.1.6. • If at least one of r1 > 1 or r2 > 1, then we may assume without loss of generality that r1 > 1. Hence b=( Q r1 i=3 [αi αi′ ] Q r2 i=1 [δi δi′ ])[β1 β2 ][β3 β4 ] · · · [βr3 −2 βr3 −1 ] [β1′ β2′ ][β3′ β4′ ] · · · [βr′ 3 −2 βr′ 3 −1 ][µ1 µ2 ][α1 α2 ][α1′ α2′ ] where µ1 , µ2 ∈ supp(a) ∩ f ix(x). Therefore, the total number of in Gxb equals {s + 1, 1}. , Chapter 3. Commuting Involution Graph of 2. Sn 55 • If r1 = r2 = 0, then n > 3(2s + 1)(this because supp(x) = 2(2s + 1) and f ix(x) > 2s + 1) and to find a path between a and x we need to consider the following: (a) If n = 3(2s + 1), then we may write x as follows x = [1 2m + 1][3 2m + 2] · · · [2m − 3 n − 1][2m − 1 n], and we set b1 = [1 3][5 7] · · · [2m − 5 2m − 3][2m+1 2m+2] · · · [n−2 n−1][2m−1 n], b2 = [1 3][5 7] · · · [2m−5 2m− 3][2 4][6 8] · · · [2m−4 2m−2][2m−1 n] and b3 = [1 3][2 4][5 6][7 8] · · · [2m− 3 2m − 2][2m + 1 2m + 2]. Thus the total number of , in x , G b1 , G b2 , G b3 receptively {s, 0}, {0, s}, {s − 1, 1}, {1, 1}. Gb1 a b2 b3 (b) If n > 3(2s + 1), then set b = [β1 β2 ][β3 β4 ] · · · [βr3 −4 βr3 −3 ][β1′ β2′ ][β3′ β4′ ] · · · [βr′ 3 −4 βr′ 3 −3 ] [µ1 µ2 ][µ3 µ4 ][µ5 µ6 ] where µ1 , µ2 , µ3 , µ4 , µ5 , µ6 ∈ supp(a) ∩ f ix(x), so that the total number , of in Gxb is {s − 1, 3}. iii- If r3 = 2s with s odd then put b=( Qt1 i=1 [αi αi′ ] Qt2 i=1 [δi δi′ ])[β1 β2 ][β3 β4 ] · · · [βr3 −1 βr3 ] [β1′ β2′ ][β3′ β4′ ] · · · [βr′ 3 −1 βr′ 3 ][µ1 µ2 ] where µ1 , µ2 ∈ supp(a) ∩ f ix(x). Here t1 = r1 − 1 , t2 = r2 if r1 6= 0 , and if r1 = 0 then set t2 = r2 − 1. The total number of , in Gxb is {s, 1}. iv- If r3 = 2s with s even then put b=( Q r1 i=1 [αi αi′ ] Q r2 i=1 [δi δi′ ])[β1 β2 ][β3 β4 ] · · · [βr3 −1 βr3 ] [β1′ β2′ ][β3′ β4′ ] · · · [βr′ 3 −1 βr′ 3 ]. Here the total number of {s, 0}. , in Gxb equals 3.6. The Diameter of C(2. Sn , X) 56 Theorem 3.6.4. Suppose that n = 2m + 2 > 10. Then DiamC(2. Sn , X) 6 4. Proof. Assume that {β1 , β2 } = f ix(a) and let x ∈ X. If f ix(a) = f ix(x) or x[β1 β2 ] = ±x, so no black vertices are connected with white vertices, by Proposition 3.6.1, we get DiamC(2. Sn , X) 6 4. Additionally, if β1x 6= β2 and {β1 , β2 } ∈ supp(x) then in a similar way as case (iii) in Theorem 2.6.3, we obtain DiamC(2. Sn , X) 6 4. Last but not least, if f ix(x) = {β1 , α} or {β2 , α} where α is not equal to neither β1 or β2 , then without loss of generality we assume that f ix(x) = {β2 , α}. To find a path between a and x, we assume that C1 , C2 , · · · , Ch are the connected components of Gx . Then all Cj ′ s are cycle or loops except one. We denote this components by Ci . Hence . Ci is a chain starting with β1 and ending with black vertices, Let ai and xi be the generators of Ci in Gx , so we may assume without loss of generality that for k > 1, xi = [2 3][4 5][6 7] · · · [2k − 2 2k − 1][1 β1 ][2k] and ai = [1 2][3 4] · · · [2k − 1 2k][β1 ]. Now let al = [β2 ] and xl = [β2 ]. Also let ail = [1 2][3 4] · · · [2k − 1 2k][β1 ][β2 ] and xil = [2 3][4 5][6 7] · · · [2k − 2 2k − 1][1 β1 ][2k][β2 ]. For k = 1 we may let ail = [1, 2][β1 ][β2 ] and xil = [1 β1 ][2][β2 ] for k = 1. Since all Cj ′ s except Ci have no white vertices connected with black vertices, by Proposition 3.6.1 it suffices to find a path between ail and xil to find a path between a and x, and for that reason we need to look at the following cases: Case I: If k = 1, then ail = [1, 2][β1 ][β2 ] and xil = [1 β1 ][2][β2 ], and Gxailil is given by . Thus in order to find the path between ail and xil , we have to consider the following cases: i- If Gx has a connected component Cj such that Cj is a cycle of length 2, we may assume that aj = [3 4][5 6] and xj = [3 5][4 6] are the corresponding parts of a and x respectively, that is they generate Cj in Gx . Now put a ilj ailj = [1 2][3 4][5 6] and xilj = [1 β1 ][3 5][4 6], so that Gxilj is given by . Now if n = 10 then we my let a = [1 2][3 4][5 6][7 8] and x = [1 9][3 5][4 6][7 8]. Taking b = [1 2][3 5][4 6][9 10] and b∗ = [1 9][3 6][4 5][2 10], we have a path from a to x. On the other hand, if Chapter 3. Commuting Involution Graph of 2. Sn 57 n > 10 it is possible to consider the following subcases: i. If Gx has connected component Cp such that Cp is a cycle of length 2, we may assume that ap = [7 8][9 10] and xp = [7 9][8 10] are the corresponding parts that generate Cp in Gx . Set ailjp = [1 2][3 4][5 6][7 8][9 10] and xiljp = [1 β1 ][3 5][4 6][7 9][8 10]. Let biljp = [1 2][3 5][4 6][7 8][β1 β2 ], a b b ∗ iljp iljp iljp b∗iljp = [1 β1 ][2 β2 ][4 6][7 8][9 10]. Then Gbiljp ,Gbiljp ∗ , Gxiljp are presented , and hence we get as d(ailjp , biljp ) = d(biljp , b∗iljp ) = d(b∗iljp , xiljp ) = 1. ii. If Gx has connected component Cp such that Cp is a cycle of length , we may assume that ap = [7 8][9 10][11 12] and 3, xp = [8 9][10 11][7 12] are the corresponding parts that generate Cp in Gx . Set ailjp = [1 2][3 4][5 6][7 8][9 10][11 12] and xiljp = [1 β1 ][3 5][4 6][8 9] [10 11][7 12]. Let biljp = [1 2][3 4][5 6][8 9][7 10][β1 β2 ],b∗iljp = [1 β1 ][2 β2 ] [3 4][5 6][7 10][11 12] and b∗∗ iljp = [1 β1 ][2 β2 ][3 5][4 6][7 11][10 12]. Then a b b iljp Gbiljp ,Gb∗iljp , Gxiljp iljp iljp b ∗∗ given by ∗ iljp while Gbiljp ∗∗ representative as follows . iii. If Gx has a connected component Cp such that Cp is a cycle of length greater than 3, let ap = [7 8][9 10] · · · [2r + 5 2r + 6] and xp = [8 9][10 11] · · · [2r + 4 2r + 5][7 2r + 6] where r > 3. Now put ailjp = [1 2][3 4][5 6][7 8] · · · [2r+5 2r+6] and xiljp = [1 β1 ][3 5][4 6][8 9][10 11] · · · a iljp is given by [2r + 4 2r + 5][7 2r + 6]. The graph Gxiljp , so that our aim is to find biljp , b∗iljp and b∗∗ iljp (if required) which connect ailjp to xiljp . For that reason we argue the following cases: • If r is even such that r = 2s and s is even, then set biljp = [1 2][3 4][5 6] [7 2r+6][8 2r+5] · · · [r+5 r+8][r+6 r+7], b∗iljp = [1 2][3 6][4 5][7 2r+ 6][8 2r+5] · · · [r+5 r+8][β1 β2 ] and b∗∗ iljp = [1 β1 ][2 β2 ][3 5][4 6][7 2r+ 6][8 2r + 5] · · · [r + 5 r + 8]. Thus the total number of , in 3.6. The Diameter of C(2. Sn , X) a b 58 b ∗ b iljp iljp iljp and Gxiljp Gbiljp , Gbiljp ∗ , Gb∗∗ iljp ∗∗ respectively {s, 0},{1, 1},{2, 0} and iljp {s − 1, 1}. • If r is even such that r = 2s and s is odd, then set biljp = [1 2][3 5][4 6][7 2r + 6][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7], b∗iljp = [1 2][3 6][4 5][7 2r + 6][8 2r + 5] · · · [r + 5 r + 8][β1 β2 ] and b∗∗ iljp = [1 β1 ][2 β2 ][3 4][5 6][7 2r + 6][8 2r + 5] · · · [r + 5 r + 8]. The total number of a b b ∗ b iljp iljp iljp , in Gbiljp and Gxiljp , Gbiljp ∗ , Gb∗∗ iljp ∗∗ iljp respectively {s + 1, 0},{1, 1},{2, 0} and {s, 1}. • If r is odd such that r = 2s + 1 and s is odd, then set biljp = [1 2][3 5][4 6][7 2r + 6][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7], b∗iljp = [1 2][3 6][4 5][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7][β1 β2 ] and b∗∗ iljp = [1 β1 ][2 β2 ][3 5][4 6][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7]. Here , the total number of a ∗ b b b iljp iljp iljp and Gxiljp in Gbiljp , Gbiljp ∗ , Gb∗∗ iljp ∗∗ iljp respectively {s + 1, 0},{1, 1},{2, 0} and {s, 1}. • If r is odd such that r = 2s + 1 and s is even,then set biljp = [1 2][3 4][5 6][7 2r + 6][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7], b∗iljp = [1 2][3 6][4 5][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7][β1 β2 ] and b∗∗ iljp = [1 β1 ][2 β2 ][3 4][5 6][8 2r + 5] · · · [r + 5 r + 8][r + 6 r + 7]. Then , the total number of a b ∗ b b iljp iljp iljp in Gbiljp , Gbiljp and Gxiljp ∗ , Gb∗∗ iljp ∗∗ iljp respectively {s, 0},{1, 1},{2, 0} and {s + 1, 1}. ii- If Gx has a connected component Cj such that Cj is a cycle of length 3, , let aj = [3 4][5 6][7 8] put ailj = [1 2][3 4][5 6][7 8] and and xj = [4 5][6 7][3 8]. Here xilj = [1 β1 ][4 5][6 7][3 8]. In this case let bilj = [β1 β2 ][3 8][5 6][4 7], b∗ilj = [1 2][β1 β2 ][4 5][6 7] and b∗∗ ilj = [2 β2 ][1 β1 ][4 7][5 6]. a b b ilj ilj ilj Then Gbilj , Gbilj ∗ and Gxilj b ∗∗ are given by ∗ ilj . On the other hand Gbilj ∗∗ is presented by . Hence by using Lemma 3.4.2 we obtain d(ailj , bilj ) = ∗∗ d(bilj , b∗ilj ) = d(b∗ilj , b∗∗ ilj ) = d(bilj , xilj ) = 1. Chapter 3. Commuting Involution Graph of 2. Sn 59 iii- If Gx has a connected component Cj such that Cj is a cycle of length more than 3, let aj = [3 4][5 6] · · · [2r + 1 2r + 2] and xj = [4 5][6 7] · · · [2r 2r + 1][3 2r + 2]. Now put ailj = [1 2][3 4][5 6][7 8] · · · [2r + 1 2r + 2] and a ilj xilj = [1 β1 ][4 5][6 7] · · · [2r 2r + 1][3 2r + 2]. The graph Gxilj is given by , so that our purpose is to find bilj , b∗ilj and b∗∗ ilj (if required) which link ailj and xilj . For that intent we look at the following cases: • If r is even such that r = 2s and s is even, then set bilj = [1 2][3 2r + 2][4 2r + 1][5 2r] · · · [r + 1 r + 4][r + 2 r + 3], b∗ilj = [1 2][β1 β1 ][4 5][2r + 1 2r][6 2r − 1] · · · [r + 2 r + 3] and b∗∗ ilj = [1 β1 ][2 β2 ][4 2r + 1][5 2r] · · · [r + a and b ∗∗ Gxilj ilj b b ilj ilj ilj , in Gbilj , Gbilj ∗ , Gb∗∗ 1 r+4][r+2 r+3]. Thus the total number of ∗ ilj respectively equals {s, 0},{1, 1},{2, 0} and {s − 1, 1}. • If r is even such that r = 2s and s is odd, then set bilj = [β1 β2 ][3 2r + 2][4 2r + 1][5 2r] · · · [r + 1 r + 4][r + 2 r + 3], b∗ilj = [1 2][β1 β1 ][4 2r][2r + 1 5][6 2r − 1] · · · [r + 2 r + 3] and b∗∗ ilj = [1 β1 ][2 β2 ][4 5][2r 2r + 1][6 2r − 1] · · · [r + 2 r + 3]. The total number of and b ∗∗ Gxilj ilj , a b b ilj ilj ilj in Gbilj , Gbilj ∗ , Gb∗∗ ∗ ilj respectively {s, 1},{1, 1},{2, 0} and {s − 2, 1}. • If r is odd such that r = 2s+1 and s is odd,then set bilj = [β1 β2 ][3 2r + 2][4 2r + 1][5 2r] · · · [r + 1 r + 4][r + 2 r + 3], b∗ilj = [1 2][β1 β1 ][4 5][2r + 1 2r][6 2r − 1] · · · [r + 2 r + 3] and b∗∗ ilj = [1 β1 ][2 β2 ][4 2r + 1][5 2r] · · · [r + 1 r + 4][r + 2 r + 3]. Here the total number of b Gxilj ilj ∗∗ b b ∗ , a ilj in Gbilj , and ilj ilj is {s, 1}, while in Gbilj respectively equals {1, 1},{2, 0}. ∗ , Gb∗∗ ilj • If r is odd such that r = 2s + 1 and s is even, then set bilj = [1 2][3 2r + 2][4 2r + 1][5 2r] · · · [r + 1 r + 4][r + 2 r + 3], b∗ilj = [1 2][β1 β1 ][4 2r][2r + 1 5][6 2r − 1] · · · [r + 2 r + 3] and b∗∗ ilj = [1 β1 ][2 β2 ][4 5][2r 2r + 1][6 2r − 1] · · · [r + 1 r + 4][r + 2 r + 3]. Thus the total number of a b b ∗ b ilj ilj ilj and Gxilj in Gbilj , Gbilj ∗ , Gb∗∗ ilj ilj 1, 1}. ∗∗ , respectively {s, 0},{1, 1},{2, 0} and {s − 3.6. The Diameter of C(2. Sn , X) 60 iv- If ω1 and ω2 be the number of cycles of length 2 and 3 and Cω1 and Cω2 be their connected component in Gx respectively. We need to consider the following sub cases: • If (ω1 , ω2 ) = (3, 0) then we suppose without loss of generality that, aω1 = [1 2][3 4][5 6][7 8][9 10][11 12][13 14] and xω1 = [1 β1 ][4 5][3 6][8 9][7 10] [12 13][11 14]. Hence if we set bω1 = [1 2][4 5][3 6][7 8][9 10][12 13][11 14], b∗ω1 = [1 2][β1 β2 ][3 6][7 8][9 10][11 12][13 14] and b∗∗ ω1 = [1 β1 ][2 β2 ][3 6] [7 8][9 10][12 13][11 14], then d(aω1 , bω1 ) = d(bω1 , b∗ω1 ) = d(b∗ω1 , b∗∗ ω1 ) = d(b∗∗ ω1 , xω1 ) = 1. aω bω bω bω Gbω 1 , Gb∗∗1 equals {2, 0}, and in Gbω 1 ∗ , Gxω11 1 , This is because the total number of ∗ ω1 1 ∗∗ in equals {1, 1}. Moreover, If (ω1 , ω2 ) equal to the one of the followings (1, 0), (0, 1), (2, 1) and (1, 2). Then by previous cases one can see immediately that there is a edges links the components of x and a. In general, if ω1 + ω2 = s and s is odd then there is always a path between a and x. This because s = 3 + r1 where r1 is an even, thus by the former cases there exist a path between the components of x and a. • If (ω1 , ω2 ) = (0, 2) then we suppose without loss of generality that, aω2 = [1 2][3 4][5 6][7 8][9 10][11 12][13 14] and xω2 = [1 β1 ][4 5][6 7][3 8][10 11] [12 13][9 14]. Hence if we set bω2 = [1 2][3 8][4 7][5 6][9 14][10 13][11 12], b∗ω2 = [1 2][β1 β2 ][4 6][5 7][9 14][10 13][11 12] and b∗∗ ω2 = [1 β1 ][2 β2 ][4 5] [6 7][9 14][10 13][11 12]. The total number of equals {2, 0}, and in bω bω ∗∗ Gb∗ 2 , Gxω22 ω2 , aω bω in Gbω 2 , Gb∗∗2 2 ∗ ω2 equals {1, 1}. We get d(aω2 , bω2 ) = ∗∗ d(bω2 , b∗ω2 ) = d(b∗ω2 , b∗∗ ω2 ) = d(bω2 , xω2 ) = 1. Additionally, if (ω1 , ω2 ) equal (2, 0) or (1, 1) then by the same argument as above we can find a path in this case. Overall, if ω1 + ω2 = s and s is even , set s = 2 + r1 where r1 in this case even. Thus the existence of the path is satisfied. Case II: If k = 2, then without loss of generality, we may set ail = [1 2][3 4][β1 ][β2 ], and xil = [2 3][1 β1 ][4][β2 ]. In order to find the path between ail and xil , we have to Chapter 3. Commuting Involution Graph of 2. Sn 61 consider the following cases: i- If Gx has a connected component Cj such that Cj is a cycle of length 2, we may assume that aj = [5 6][7 8] and xj = [6 7][8 5] are the corresponding parts of a and x respectively, that is they generate Cj in Gx . Now put ailj = [1 2][3 4][5 6][7 8] and xilj = [2 3][1 β1 ][6 7][8 5], and let bilj = [1 4][2 3][5 8][6 7], b∗ilj = [1 4][β1 β2 ][5 6][7 8] a and ∗ b ilj ilj [1 β1 ][4 β2 ][5 7][6 8]. Thus Gbilj , Gbilj ∗∗ are given by b b iljp iljp whilst Gbiljp ∗ , Gxiljp ∗∗ b∗∗ ilj = , . So we obtain are represented as ∗∗ d(ailj , bilj ) = d(bilj , b∗ilj ) = d(b∗ilj , b∗∗ ilj ) = d(bilj , xilj ) = 1. ii- If Gx has a connected component Cj such that Cj is a cycle of length 3, , then let aj = [5 6][7 8][9 10] and xj = [6 7][8 9][10 5]. Here put ailj = [1 2][3 4][5 6][7 8][9 10] and xilj = [2 3][1 β1 ][6 7][8 9][10 5]. In this case let bilj = [1 4][2 3][5 6][7 10][8 9] , b∗ilj = [1 4][β1 β2 ][5 6][7 9][8 10] and a b ∗ ilj ilj b∗∗ ilj = [4 β2 ][1 β1 ][5 6][8 9][7 10]. Therefore, Gbilj , Gbilj ∗∗ are given by b b , while Gb∗ilj , Gxilj ilj ilj ∗∗ are represented by . This yields d(ailj , bilj ) = d(bilj , b∗ilj ) = d(b∗ilj , b∗∗ ilj ) = d(b∗∗ ilj , xilj ) = 1. iii- If Gx has a connected component Cj such that Cj is a cycle of length greater than 3, then without loss of generality let aj = [5 6][7 8] · · · [2r + 3 2r + 4] and xj = [6 7][8 9] · · · [5 2r + 4] where r > 3. Also let ailj = [1 2][3 4][5 6][7 8] · · · [2r + 3 2r + 4] and xilj = [2 3][1 β1 ][6 7][8 9] · · · [2r + a ilj 2 2r + 3][5 2r + 4]. Then the graph Gxilj is given by , so that our purpose is to find bilj , b∗ilj and b∗∗ ilj which form a path between ailj and xilj . For that we check the following cases: • If r is even such that r = 2s and s is even, then set bilj = [2 3][1 4][5 6][7 2r +4][8 2r + 3] · · · [r + 4 r + 7][r + 5 r + 6], b∗ilj = [1 4][β1 β2 ][5 7][6 2r + 4][8 2r + 3] · · · [r + 4 r + 7][r + 5 r + 6] and b∗∗ ilj = [4 β2 ][1 β1 ][6 7][5 2r + 3.6. The Diameter of C(2. Sn , X) 62 4][8 2r + 3] · · · [r + 4 r + 7][r + 5 r + 6]. Then the total number of a b b ∗ b ilj ilj ilj and Gxilj , in Gbilj , Gbilj ∗ , Gb∗∗ ilj ∗∗ ilj respectively {s, 0},{1, 1},{2, 0} and {s − 1, 1}. • If r is even such that r = 2s and s is odd, then let bilj = [2 3][1 4][5 r + 5][6 r + 6][7 2r + 4][8 2r + 3] · · · [r + 3 r + 8][r + 4 r + 7], b∗ilj = [1 4][β1 β2 ][5 r + 6][6 r + 5][7 2r + 4][8 2r + 3] · · · [r + 3 r + 8][r + 4 r + 7] and bilj ∗∗ = [4 β2 ][1 β1 ][5 6][7 2r +4][8 2r +3] · · · [r +4 r +7][r +5 r +6]. , Thus the total number of a b b ∗ b ilj ilj ilj in Gbilj , Gbilj and Gxilj ∗ , Gb∗∗ ilj ∗∗ ilj re- spectively {s + 1, 0},{1, 1},{2, 0} and {s, 1}. • If r is odd such that r = 2s + 1 and s is odd, then let bilj = [2 3][1 4][5 6][7 2r+4][8 2r+3] · · · [r+3 r+8][r+4 r+7][r+5 r+6], b∗ilj = [1 4][β1 β2 ][5 7][6 2r + 4][8 2r + 3] · · · [r + 3 r + 8][r + 4 r + 7][r + 5 r + 6] and b∗∗ ilj = [4 β2 ][1 β1][5 6][7 2r + 4][8 2r + 3] · · · [r + 4 r + 7][r + 5 r + 6]. a b ∗ b b ilj ilj ilj and Gxilj , in Gbilj , Gbilj ∗ , Gb∗∗ ilj Therefore, the total number of ∗∗ ilj respectively {s + 1, 0},{1, 1},{2, 0} and {s, 1}. • If r is odd such that r = 2s + 1 and s is even, then let bilj = [1 2][3 4][5 6][7 2r + 4][8 2r + 3] · · · [r + 4 r + 7][r + 5 r + 6], b∗ilj = [1 2][3 4][5 6][7 2r + 3][8 2r + 4][9 2r + 2] · · · [r + 4 r + 7][β1 β2 ] and b∗∗ ilj = [1 2][3 β1 ][4 β2 ][5 6][7 2r + 4][8 2r + 3][9 2r + 2] · · · [r + 4 r + 7]. The total number of , a b b ∗ b ilj ilj ilj in Gbilj and Gxilj , Gbilj ∗ , Gb∗∗ ilj ilj ∗∗ respec- tively {s, 0},{1, 1},{2, 0} and {s + 1, 1}. iv- If ω1 and ω2 are the number of cycles of length 2 and 3 respectively, then we need to consider the following sub cases: • If ω1 = ω2 = 1 then we suppose without loss of generality that Cω1 and Cω2 are the corresponding connected components in Gx . Let aω1 = [5 6][7 8] and aω2 = [9 10][11 12][13 14]. Also let xω1 = [6 7][5 8] and xω2 = [10 11][12 13][9 14]. Hence set ailω1 ω2 = [1 2][3 4][5 6][7 8][9 10][11 12][13 14][β1 ][β2 ] and xilω1 ω2 = [2 3][1 β1 ][6 7][5 8][10 11][12 13][9 14][4] [β2 ], we may set ξ = ilω1 ω2 . Set bξ = [2 3][1 4][5 6][7 8][9 10][12 13][11 14], Chapter 3. Commuting Involution Graph of 2. Sn 63 b∗ξ = [β1 β2 ][1 4][5 7][6 8][9 10][12 13][11 14] and b∗∗ ξ = [1 β1 ][4 β2 ][6 7][5 8] [9 10][12 13][11 14]. Since the total number of equals {2, 0}, and in b b ∗∗ Gb∗ξ , Gxξξ ξ a b ∗ ξ in Gbξξ , Gb∗∗ , ξ equals {1, 1}, we obtain that d(aξ , bξ ) = ∗∗ d(bξ , b∗ξ ) = d(b∗ξ , b∗∗ ξ ) = d(bξ , xξ ) = 1. • If ω1 = 0 and ω2 = 2 or vice versa, we may assume without loss of generality that ω1 = 2 and ω2 = 0 (if ω1 = 0 and ω2 = 2 then by the same way a path between x and b can be found). Then suppose that Cω1 and Cω1 ∗ are the connected components in Gx , with component parts aω1 = [5 6][7 8] , aω1∗ = [9 10][11 12], xω1 = [6 7][5 8] and xω1∗ = [10 11][9 12]. Thus if we let ailω1 ω1∗ = [1 2][3 4][5 6][7 8][9 10][11 12] and xilω1 ω1∗ = [2 3][1 β1 ][6 7][5 8][10 11][9 12], we may let ξ = ilω1 ω1∗ . Put bξ = [2 3][1 4][5 7][6 8][9 10][11 12], b∗ξ = [β1 β2 ][1 4][5 7][6 8][10 11][9 12] and b∗∗ ξ = [1 β1 ][4 β2 ][5 6][7 8][10 11][9 12]. We get that d(aξ , bξ ) = d(bξ , b∗ξ ) = d(b∗ξ , b∗ξ ) = d(b∗∗ ξ , xξ ) = 1, as one can see that the total number of , a b ∗ b b ξ in Gbξξ , Gb∗∗ equals {2, 0}, and in Gb∗ξ , Gxξξ ξ ξ ∗∗ equals {1, 1}. • If ω1 + ω2 = s and s odd, then there is a path in this case. This because if s = 1 then by previous cases the path exists and since s = r + 1 where r is even and again the path exists by the Case 4.1. Whilst if s is even, then if s = 2 by the above there is a path between a and x. Moreover, if s > 2, then s = 2 + q, such that q is an even and again by the above yields the existence of a path between the components of a and b such that d(a, x) 6 4. Case III: If k > 2, then Gxailil is given by . Our aim is to find bil , b∗il and b∗∗ il (if required) which build a path between ail and xil . For that purpose we need to take into account the following cases: i- If k = 2s and s is even, then if s = 2 we let bil = [β1 β2 ][3 8][4 7][5 6], b∗il = [3 β1 ][8 β2 ][4 5][6 7] and b∗∗ il = [3 β1 ][12][4 7][5 6] . Otherwise take 3.6. The Diameter of C(2. Sn , X) 64 bil = [β1 β2 ][3 2k][4 2k − 1][5 2k − 2] · · · [k − 1 k + 4][k k + 3][k + 1 k + 2] , b∗il = [2k β2 ][3 β1 ][4 2k − 1][5 2k − 2] · · · [k − 1 k + 4][k k + 1][k + 2 k + 3] and b∗∗ il = [1 2][3 β1 ][4 2k − 1][5 2k − 2] · · · [k − 1 k + 4][k k + 3][k + 1 k + 2]. ∗ il and Gxbilil , in Gbailil , Gbbilil∗ , Gbb∗∗ Thus the total number of ∗∗ il respectively {s − 1, 1},{2, 0}, {1, 1} and {s, 0}. ii- If k = 2s and s is odd, then if s = 3 set bil = [β1 β2 ][3 12][4 11] [5 6][9 10][7 8], b∗il = [12 β2 ][3 β1 ][4 11][5 10][6 9][7 8] and b∗∗ il = [1 2][3 β1 ][4 11][5 10][6 7] [8 9] . Otherwise take bil = [β1 β2 ][3 2k][4 2k − 1][5 2k − 2] · · · [k − 2 k + 5][k − 1 k][k + 3 k + 4][k + 1 k + 2], b∗il = [2k β2 ][3 β1 ][4 2k − 1][5 2k − 2][6 2k − 3] · · · [k − 2 k + 5][k − 1 k + 4][k k + 3][k + 1 k + 2] and b∗∗ il = [1 2][3 β1 ][4 2k −1][5 2k −2] · · · [k −1 k +4][k k +1][k +2 k +3]. One can show ∗ il , in Gbailil , Gbbilil∗ , Gbb∗∗ and Gxbilil that the total number of il ∗∗ respectively {s − 2, 1},{2, 0}, {1, 1} and {s − 1, 0}. iii- If k = 2s + 1 and s is odd, then if s = 1 let bil = [β1 β2 ][3 6][4 5] and b∗il = [1 2][3 β1 ][6 β2 ] . Otherwise take bil = [k + 1 k + 2][3 2k][4 2k − 1][5 2k − 2] · · · [k k + 3][β1 β2 ] and b∗il = [1 2][3 β1 ][4 2k − 1][5 2k − 2][6 2k − 3] · · · [k k+3][2k β2 ] . So that that the total number of , in Gbailil , Gxbilil ∗ equals {s, 1}, while in Gbb∗il it equals {1, 1}. il iv- If k = 2s + 1 and s is even, then if s = 2 let bil = [β1 β2 ][3 10][4 9] [5 6][7 8], b∗il = [3 β1 ][10 β2 ][4 9][5 7][6 8] and b∗∗ il = [1 2][3 β1 ][4 9] [5 8][6 7] . Otherwise take bil = [β1 β2 ][3 2k][4 2k − 1][5 2k − 2] · · · [k − 1 k + 4][k k + 1][k + 2 k + 3], b∗il = [2k β2 ][3 β1 ][4 2k − 1][5 2k − 2][6 2k − 3] · · · [k − 1 k + 4][k k +2][k +1 k +3] and b∗∗ il = [1 2][3 β1 ][4 2k −1][5 2k −2][6 2k −3] · · · [k − 1 k + 4][k k + 3][k + 1 k + 2]. We can show that the total number of ∗ il and Gxbilil in Gbailil , Gbbilil∗ , Gbb∗∗ il ∗∗ , respectively equals {s − 1, 1},{2, 0}, {1, 1} and {s, 0}. Thus we conclude from the above cases and Proposition 3.6.1, that d(a, x) 6 4. Chapter 3. Commuting Involution Graph of 2. Sn 65 Nevertheless, Theorem 3.6.3 is not satisfied when n = 8 and m = 3. Using Magma the analysis of the discs structure of t is as follows: 1 t 10 10 13 ∆2 (t) ∆1 (t) 1 1 2 8 ∆3 (t) 3 2 2 ∆4 (t) 5 2 ∆5 (t) 3 6 10 Where |∆1 (t)| = 13, |∆2 (t)| = 60, |∆3 (t)| = 294, |∆3 (t)| = 424 and |∆5 (t) = 48. Furthermore, for xi ∈ ∆i (t), i = 1, 2, 3, 4, 5, we calculate the following: The number of elements in ∆1 (t) which commute with t, x1 , x2 respectively equals 13, 2, 1. The number of elements in ∆2 (t) which commute with x1 , x2 , x3 is 10, 2, 3 respectively. The number of elements in ∆3 (t) which commute with x2 , x3 , x4 is 10, 2, 5 respectively. Whilst the number of elements in ∆4 (t) which commute with x3 , x4 , x5 respectively equals 8, 6, 3. Finally, the number of elements in ∆5 (t) which commute with x4 , x5 respectively equals 2, 10. As we can see from the above calculation the DiamC(2. S8 , X) = 5, where t = . [1 2][3 4][5 6] and X = t2 S8 . However, Theorem 3.6.3 shows that DiamC(2. Sn , X) 6 4 for n = 2m + 2 > 10. Chapter 4 Commuting Involution Graphs of Double Covers of Sporadic Groups and Their Automorphism Groups 4.1 Introduction Suppose that G is a sporadic simple group and X a conjugacy class of involutions in G. The commuting involution graphs C(G, X) and C(G.2, X) have been studied by P.Rowley and his PhD student (Chris Bates, David Bundy and Sarah Hart) in [16]. In this chapter we consider the commuting involution graphs of double covers of sporadic simple groups C(2. G, X), and double covers of their automorphism groups C(2. G.2, X). One can see from the classification of finite simple groups that sporadic simple groups are as follows: • M11 , M12 , M22 , M23 , M24 (the Mathieu groups). • Co1 , Co2 , Co3 , M cL, HS, Suz, J2 (the Leech lattice groups). • F i22 , F i23 , F i′24 (the Fischer groups). 66 Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) 67 • M, B, T h, HN, He (the Monstrous groups). • J1 , J3 , J4 , O′ N, Ru, Ly (the pariahs). In most of the results that we deal with in this chapter, computational technical methods are used to investigate the commuting involution graphs. Magma [23] and GAP [41] are used to analyze the disc structure of the graphs. Now, for a fixed involution t in G, the disc △i (t) is a union of CG (t)-orbits of tG (as CG (t) acts by conjugation on tG ). Therefore, our aim is to determine these orbits and check which of these lie in △i (t). 4.2 General Results Our strategy to obtain a complete set of representatives for the CG (t)-orbits on X is to consider an arbitrary group element g ∈ G and use a variety of computational algorithms and group theory results to obtain the complete set of representatives. Suppose x ∈ X. Then the size of the orbit xCG (t) is equal to [CG (t) : CCG (t) (x)]. However, the computational code “IsConjugate” will be employed in many cases to determine whether an element y ∈ X belongs to xCG (t) . The following lemma is essential in determining the discs of the commuting involution graph C(G, X). Lemma 4.2.1. For a finite group G, each disc ∆i (t) of the commuting involution graph C(G, X) is a union of particular CG (t)-orbits of X, (where CG (t) acts on X by conjugation and t ∈ X). Proof. It is enough to prove that for x ∈ ∆i (t) and z ∈ CG (t), xz ∈ ∆i (t). As d(t, x) = i, the shortest path between t, x is of length i. Moreover, if we conjugate the element of this path by z, since z ∈ CG (t) we get another path but this time between t, xz of length i. If d(t, xz ) < i then conjugating by z −1 we obtain d(t, x) < i, a contradiction. Therefore xz ∈ ∆i (t). The next theorem (mentioned in [16]) gives basic results which we use as an important tool to determine the elements of ∆i (t). 4.2. General Results 68 Theorem 4.2.2. Let G be a finite group, t ∈ G an involution and let X = tG the conjugacy class of t. suppose that x ∈ X and put z = tx, and let m equal the order of z. The following holds 1. CCG (t) (x) = CG (t) ∩ CG (x) = CCG (z) (t) = CCG (z) (x); 2. m = 2 if and only if x ∈ ∆1 (t); 3. x ∈ ∆2 (t) if m is even, m > 4 and z m/2 ∈ X; 4. d(x, t) > 3 if CCG (z) (x) ∩ X = ∅. Specially, if the order of CCG (z) (x) is odd then d(x, t) > 3; 5. If m is odd and there are no elements g ∈ G with the property that the order of g is 2m, g 2 = z and g m ∈ X, then d(x, t) > 3. Proof. Full proof can be found in [16]. Definition 4.2.3. [55] Let G is a group act on a finite set Ω, for g ∈ G define χ(g) = |{α ∈ Ω|αg = α}|. The non-negative integer value function χ is called the Permutation Character associated with the action. The number of CG (t)-orbits under the action of conjugation on tG can be calculated by using the character table of the group, as we can seen in the following result: Proposition 4.2.4. Let G be a group acting transitively on a finite set Ω, with a permutation character χ. Suppose that α ∈ Ω and that Gα has exactly k orbits on Ω. Then hχ, χi = k. Proof. See [55, Corollary 5.15]. The quantity k in Proposition 4.2.3 is called the permutation rank of Gα on Ω. Therefore, the permutation rank of CG (t) on X is equal to the number of CG (t)-orbits under the conjugation action on X. Assume that C is a G-conjugacy class. It is obvious that the set XC = {x ∈ X : tx ∈ C} under the conjugation action of CG (t) breaks up into suborbits. Hence to Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) 69 determine all the suborbits of X, we have to identify the CG (t)-orbits of XC , for all those C such that XC 6= φ. The next definition gives us the size of the set XC , and therefore, the size of suborbits of XC : Definition 4.2.5. Let Ci , Cj and Ck be conjugacy classes of a finite group G. Then for a fixed element g ∈ Ck , define the set aijk = |{(gi , gj ) ∈ Ci × Cj | gi gj = g}|. Then for all possible i, j, k the value aijk is called a class class structure constants for G. The next lemma will be used to compute the structure constant for G. Lemma 4.2.6. [44] Let G be a finite group with n conjugacy classes C1 , C2 , . . . Cn . Then for all i, j, k we have aijk = P χ(gi )χ(gj )χ(gk ) |G| χ∈Irr(G) |CG (gi )||CG (gj )| χ(1) where gi , gj and gj are respectively in Ci , Cj and Ck . Proof. Full proof can be found in [44, Lemma 2.12]. Now, since |XC | = |{(c, x) ∈ C × X | cx = t}|, by employing Lemma 4.2.5 we get |XC | = P χ(g)χ(t)2 |G| |CG (g)||CG (t)| χ∈Irr(G) χ(1) Therefore, from the complex character table of G, which is available in GAP character table library, and using the GAP function “Class Multiplication Coefficient” we immediately obtain the size of XC . Indeed, at the end of this thesis we provide Magma code for finding |XC |. The following lemma illustrates how we can employ the known representative of a CG (t)-orbit to get a new one. Lemma 4.2.7. [65] Let x ∈ XC , so that z = tx ∈ C and suppose n is a divisor of the order of z. Let D be the conjugacy class containing z n . We define the element x(n) = tz n and note the following properties: 4.2. General Results 70 i- x(n) ∈ XD ; ii- if x, y ∈ X are CG (t)-conjugate then so are x(n) and y (n) ; and iii- all CG (t)-conjugates of x(n) arise as y (n) for some CG (t)-conjugate y of x. Proof. Full proof can be seen in [65]. Let G be a finite group whose a centre is a cyclic group Z = hzi of order 2. If θ : G −→ G/Z is the natural homomorphism then we see from Theorem 3.3.1 that for any conjugacy class C̄ of Ḡ = G/Z, the inverse image θ−1 (C̄) is either a conjugacy class of G or a union of two classes in G. Moreover, the second case appears exactly when there is no element of θ−1 (C̄) conjugate to z times itself. Now, under the above conditions one can show the following: Lemma 4.2.8. Let C̄ be a conjugacy class of involutions in Ḡ = G/Z, and suppose that θ−1 (C̄) = C1 ∪ C2 where C1 , C2 are conjugacy class of involutions in G. Then C(G, C) ∼ = C(Ḡ, C¯1 ) ∼ = C(Ḡ, C¯2 ). Proof. We only give the proof for the class C¯1 . The class C¯2 is proved in a similar way. For a ∈ C¯1 we only need to show that x ∈ ∆1 (a) if and only if x̄ ∈ ∆1 (ā) where ā and x̄ are the lifting of a, x in C¯1 respectively. The covering map implies that if d(a, x) = 1 then d(ā, x̄) = 1. Conversely assume that d(ā, x̄) = 1. If d(a, x) 6= 1, we have [a, x] = z, but this mean that a is conjugate to za which contradicts Theorem 3.3.1. Hence we must have d(a, x) = 1. Thus we get our result. Our main target is to determine the commuting involution graph for all involution classes X̄ of sporadic groups and their automorphism groups, with the condition that the inverse image of X̄ under the covering map (which we call it X) is equal to the union of involution classes in the double cover of the sporadic group or their automorphism groups. Otherwise we will end up with same situation as in Lemma 4.2.7. The following table provides us the complete set of conjugacy classes of involutions, which satisfy the above conditions with their permutation ranks. Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) 71 Table 4.1: Disc sizes and Permutation Character Group 2.M12 .2 2.M22 .2 2.Suz.2 2.Hs.2 2.Hs.2 2.Co1 2.F i22 2.F i22 .2 2.F i22 .2 2.F i22 .2 Class 2D 2F 2D 2B 2E 2D 2F 2B 2C 2F Size of Class 1584 2772 4717440 2200 46200 21361158000 72972900 7020 123552 38918880 Permutation Rank 27 28 115 8 63 465 438 6 7 106 For the remainder of this chapter we give information regarding the computational techniques involved in each case. We also give tables of the sizes of CG (t)-orbits inside the conjugacy classes of G, which means the size of sets ∆i (t) ∩ XC , where C is a G-conjugacy class. We should also refer the reader to [71] for general group-theoretic definitions and results, related to these groups. 4.3 The Mathieu Groups In the 19th century Emile Mathieu discovered the Mathieu groups which are the first family of sporadic simple groups (see [50, 51]). The largest Mathieu group, M24 , may be defined as an automorphism group of the Steiner system S(5, 8, 24), while the groups M23 and M22 can be defined to be the point and pointwise stabilizer subgroups in M24 respectively. The groupM12 is a subgroup of M24 and stabilizes a dodecad(a 12-elements subset) of S(5, 8, 24). Finally, the group M11 is defined as a point Stabilizer subgroup of M12 . The mathematical tool that may be used to deal with M24 is the Miracle Octad Generator. For more information about the MOG see [33]. According to the Table 4.1 we are interested in the cases of (2. M12 .2 , 2D) and (2. M22 .2 , 2F ). Now we outline the procedure of finding the ∆i (t) ∩ XC for those groups. To do so we create the following algorithm to finding suborbit representatives and decide which one of these 4.3. The Mathieu Groups 72 representatives lie in ∆i (t). The structure of this algorithm depends on the ability to calculate CG (t)-orbits and it is summarized as follows : Algorithm 2 . Input: G is either 2.M 12.2 or 2.M 22.2, t an involution in 2D or 2F respectively; i: r −→ Random(tG {t}), set Reps −→ {r} ii: f or x ∈ tG {t} check if and CoR −→ rCG (t) . x ∈ / CoR, then CoR 7−→ CoR ∪ {xCG (t) }; and Reps 7−→ Reps ∪ {x}. iii: for y in Reps do: if y commutes with t then ∆1 (t) 7−→ ∆1 (t) ∪ {y CG (t) }. iv: for y in Reps{∆i−1 (t) ∪ ∆i−2 (t) ∪ · · · ∆i−(i−1) (t)}, where ∆0 (t) = {t} do: if there exist y1 ∈ ∆i−1 (t) ∩ CG (y) then ∆i (t) 7−→ ∆i (t) ∪ {y CG (t) }. v: if |{∆1 (t) ∪ ∆2 (t) ∪ · · · ∆(j) (t)}| = |G| |CG (t)| , then the diameter of the graph is equal to j, where j ∈ N. vi: for x ∈ ∆i (t) do: if tx ∈ C then x ∈ XC for G-conjugacy class C. Output: the set of suborbit representatives and ∆i (t) structure. Now we employ Algorithm 2 to get the structure of C(G, X), when G is either 2.M 12.2 or 2.M 22.2 and t is an involution in 2D or 2F respectively. In the following tables we give the size of suborbits of CG (t)-orbits on X = tG inside the classes of G intersection ∆i (t): Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) 73 Table 4.2: C(2. M12 .2, 2D) Class 2A 2BC 3A 3B 4A 5B 6A 6B 6CD 10A 11A 12A 20A 22A ∆1 (t) 1 15 − − − − − − − − − − − − ∆2 (t) − − − 60 − 60, 60 − 60 − 60, 60 − − − − ∆3 (t) − − − − − − − − 60, 60 − 120, 120 − 120, 120 120, 120 ∆4 (t) − − 20, 20 − 30 − 20, 20 − − − − 120 − − ∆5 (t) − − − − 2 − − − − − − − − − Table 4.3: C(2. M22 .2, 2F ) Class 2A 2DE 3A 4CD 4F 5A 6A 6BC 10A 11A 22A 4.4 ∆1 (t) 1 5, 20 − − − − − − − − − ∆2 (t) − − − − 40, 40, 40, 40 160, 160 − − 160, 160 − − ∆3 (t) − − 40, 40 80 − − 40, 40 80, 80 − 320, 320 320, 320 Leech Lattice Groups The Leech Lattice is closely related to the Golay code, which is defined as follows. Let Λ be a set of integral vectors (a1 , · · · , a24 ) such that: • for i ∈ {1, 2, · · · , 24}, ai ≡ m mod 2; 4.4. Leech Lattice Groups 74 • The set {i : ai ≡ h mod 4} is in the Golay code, for each h, and • P24 i=1 ai ≡ 4m mod 8. A group of 24-dimensional matrices will be the automorphism group of Λ, which has central involution denoted by I24 . When we factor I24 out of the automorphism group 1 P24 2 a . of Λ we get a simple group, Co1 . The Leech Lattice has a norm defined by 8 i=1 i Therefore, the Conway groups Co2 and Co3 are defined by the Stabilizer of a vector in Co1 with norm 4 and 6 respectively. To find the commuting involution graphs for the double cover of the automorphism group of the Higman-Sims group, HS, and the Suzuki group Suz we will use Algorithm 2. For the double cover of the Conway group Co1 the situation is more complicated since the permutation rank in this case is 465, and the permutation representation has 196560 points. This means, of course, the inability to find the full CG (t)-orbits. For that reason we are going to build the next algorithm which can be realised by appealing to Lemma 4.2.5 and Lemma 4.2.6 and the fact that ∆i (t) is a union of specific CG (t)-orbits see (Lemma 4.2.1). Also parts 1 and 4 in Theorem 4.2.2 will be used in this algorithm to give the sizes of the suborbits in ∆i (t). Algorithm 3 . Input: G is either 2.Co1 , t an involution in 2D; i- for j ∈ {1, 2, · · · , ♯Classes(G)}; set ST Cj −→ 0; Repj −→ {},Xj −→ 0. ii- there is k ∈ {1, 2, · · · , ♯Classes(G)} such that t is conjugate to Classes(G)[k, 3]. iii- for i := 1 to ♯Classes(G) do ST Cj −→ ST Cj + (CharacterT able(G)[i, j] ∗ CharacterT able(G)[i, k] ∗ Conjugate(CharacterT able(G)[i, k])) /(CharacterT able(G)[i, 1]). iv- ST Cj 7−→ ST Cj ∗ Classes(G)[j, 2]/♯CG (t). Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) 75 v- for x 7−→ Random(tG ) do if t ∗ x and Classes(G)[j, 3] conjugate in G, then vi- Repj 7−→ Repj ∪ {x}; and Xj = Xj + (♯CG (t)/♯CCG (t) (x)), if ST Cj Xj then Stop; equal else; vii- pick y 7−→ Random(tG ) such that t ∗ y and Classes(G)[j, 3] conjugate in G, and no involutions in Repj are conjugate with y in CG (t) . viii- repeat the steps[v,vi] until the step v holds. ix- set N ewRep −→ {} , for x ∈ Repj and m ∈ divisor(order(t ∗ x)) do: N ewRep 7−→ N ewRep ∪ {t ∗ (t ∗ x)m } x- check the elements of N ewRep to see if they are in the same CG (t)-orbit, by using IsConjgate Magma code. xi- put REP to be the set of all CG (t)−orbit representatives, then Rep(∆1 (t)) 7−→ REP ∩ CG (t). xii- for w ∈ REP Rep(∆1 (t)) check if CCG (t) (w) has an involution in tG then w ∈ ∆2 (t); else; xiii- search for w1 ∈ ∆1 (t), such that ∆1 (w1 ) ∩ ∆1 (w) 6= φ, therefore w ∈ ∆3 (t). Output: the set of suborbit representative, ∆i (t) structure. Now we utilise Algorithms 1 and 2 to obtain the disc structure of C(G, X), when G is one of our aforementioned Leech Lattice Groups. In the following tables we give the size of suborbits of CG (t)-orbits on X = tG inside the classes of G intersection ∆i (t): • The first three tables deal with the groups 2. HS.2 and 2. Suz.2. In each case the CG (t)-orbits are calculated and the disc structure is analyzed. 4.4. Leech Lattice Groups 76 Table 4.4: C(2. HS.2, 2B) Class 2A 2CD 3A 4B 4D 6A ∆1 (t) 1 105 − − − − ∆2 (t) − − 336 − 1260 336 ∆3 (t) − − − 56 − − Table 4.5: C(2. HS.2, 2E) Class 2A 2CD 3A 4A 4B 4D 4F 5A 5B 5C 6A 6CD 7A 8D 10A 10B 10C 10DE 11A 12A 12B 14A 15A 20A 20B 20G 22A 30A ∆1 (t) 1 5, 60 − − − − − − − − − − − − − − − − − − − − − − − − − − ∆2 (t) − − 160 − − 60 240, 240 − − 960, 960 160 480 − − − − 960, 960 − − − − − − − − − − − ∆3 (t) − − 240 − 120 240, 240 − − 320 − 240 480 1920 480, 480, 480, 480 − 320 − 960, 960 1920, 1920 960, 960 960, 960, 960, 960 1920 1920 960, 960, 960, 960 − 960, 960, 960, 960 1920, 1920 1920 ∆4 (t) − − − 240 − 240 − 192, 192 − − − − − − 192, 192 − − − − − − − − − 1920 − − − Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) 77 Table 4.6: C(2. Suz.2, 2D) Class 2A 2BC 3A 3B 3C 4B 4CD 4E 4G 5A 5B 6A 6B 6CD 6E 6F G 6HI 7A 8KM 9A 10A 10B 10CD 11A 12BC 12D 12F 12H 12I 12K 14A 15B 18A 18B 18C 20EF G 22B 28A 30B ∆1 (t) 1 495 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ∆2 (t) − − − 2640, 2640 2640 − − − 23760 − 47520 − 2640, 2640 − 2640 − 23760, 23760 − − − − 47520 − − − − − − − − − − − − − − − − − ∆3 (t) − − − − − − 2970 23760 − − − − − 5940, 5940 − 15840, 15840, 15840, 15840 15840 47520 − 95040, 95040, 31680, 31680 − − 47520, 47520 190080 23760, 23760 47520, 47520 47520, 47520 − 47520, 47520, 47520, 47520 47520, 47520, 47520, 47520 47520 95040, 95040 95040, 95040, 31680, 31680 95040, 95040, 95040, 95040 95040, 95040, 95040, 95040 95040, 95040 190080 190080 95040, 95040 ∆4 (t) − − 132, 132 1760 − 1584 990 − − 9504, 9504 − 132, 132 1760 3960, 3960 − − − − 47520, 4750 − 9504, 9504 − − − − 15840, 15840 15840, 15840 31680 − 47520, 47520, 47520, 47520 − − − − − − − − − 4.4. Leech Lattice Groups 78 • In the following table we compute the CG (t)-orbit sizes and their position in ∆i (t) for the commuting involution graph of 2. Co1 , with the involution class 2D. Therefore, the structure of the graph is determined. Table 4.7: C(2. Co1 , 2D) Class 2A 2BC 2D 3A 3B 3C 3D 4A 4BC 4DE ∆1 (t) 1 7920, 495 95040, 1584, 25344 − − − − − − − 4F − 4G − 4H − 5A 5B 5C 6A 6B 6C 6DE − − − − − − − ∆2 (t) − − − − 135168, 135168, 112640 − 2027520 50688 15840, 126720 1520640,95040,95040,380160, 380160,760320 190080,190080,380160,3041280, 760320 190080,380160,380160,3041280, 253440,253440,253440,253440 1013760,1013760,6082560, 6082560 − 1622016, 1622016, 12165120 − − 135168, 135168, 112640 − − ∆3 (t) − − − 2048 − 450560, 450560 − − − − − − − 1622016 − 9732096, 9732096 2048 − 450560, 450560 1013760 Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) Class 6F 6GH 6IJ ∆1 (t) − − − 6KL − 6M N 6O − − 6P 7A 7B 8A 8CD − − − − − 8E 8F − − 8GH − 9A 9BC − − 10A 10B 10C 10DE 10F − − − − − 10G 10H − − 10I − ∆2 (t) 2027520 − 2027520,2027520,2027520, 2027520 2027520,2027520,2027520, 2027520,6082560,6082560, 1013760,4055040 − 6082560,6082560,6082560, 6082560,6082560,6082560, 6082560,6082560 24330240, 24330240 − − − 12165120,12165120,6082560, 6082560 12165120, 12165120 12165120,12165120,12165120, 12165120,12165120 6082560,6082560,6082560, 6082560,3041280,3041280, 3041280,3041280,12165120, 12165120,12165120,12165120 − − − 12165120,1622016, 1622016 − − 24330240,24330240,24330240, 24330240 − 24330240,24330240,24330240, 24330240 − ∆3 (t) − 4055040, 4055040 − − 16220160, 16220160 − − 16220160 32440320, 32440320 2027520, 2027520 − − 12165120,12165120, 12165120,12165120 − 32440320, 32440320 32440320,32440320,32440320, 32440320 1622016 − 9732096, 9732096 24330240 − 48660480, 48660480 − 48660480, 48660480 79 4.4. Leech Lattice Groups Class 10J ∆1 (t) − ∆2 (t) 24330240,24330240,24330240, 24330240,24330240,24330240, 48660480 − − − − − 11A 12A 12BC 12D 12EF − − − − − 12G 12HI − − 12J 12KL − − 12M − 12N O − 12P Q − 12R − 12S 13A 14A 14B 14CD 15A 15BC 15D − − − − − − − − 24330240,24330240,24330240, 24330240,12165120,12165120, 12165120,12165120 24330240,24330240,24330240, 24330240,24330240,24330240, 24330240,24330240,8110080, 8110080,8110080,8110080 97320960, 97320960 − − − − − − − 15E − − 16AB − − 18A − − − 24330240,24330240,8110080, 8110080 48660480 − 24330240,24330240,24330240, 24330240,24330240,24330240, 24330240,24330240 − 80 ∆3 (t) − 194641920 3244032 1013760, 1013760, 6082560 24330240, 24330240 16220160,16220160,16220160, 16220160 48660480 8110080 − 48660480,48660480,24330240, 24330240 − 48660480,48660480,48660480, 48660480 − − − 97320960 16220160 32440320, 32440320 97320960, 97320960 19464192, 19464192 97320960 97320960, 97320960, 32440320, 32440320 97320960, 97320960, 97320960, 97320960 48660480, 48660480, 48660480, 48660480 32440320, 32440320 Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) Class 18BC 18DE 18F GHI 20A 20B 20CD 20E 20F ∆1 (t) − − − − − − − − 20G 21A 21B 21C 22A 24CD 24E 24HIJ 26A 28A 28BC 30A 30BC 30DE 30F 30GHIJK 33A 35A 36AB 42A 42B 42C 52A 60AB 60CD 66A 70A 84A − − − − − − − − − − − − − − − − − − − − − − − − − − − − ∆2 (t) − − − 48660480, 48660480 − − − 97320960,97320960, 97320960,97320960 − − − − − − − − − − − − − − − − − − − − − − − − − − − − 81 ∆3 (t) 32440320, 32440320, 32440320, 32440320 97320960, 97320960 97320960, 97320960, 97320960, 97320960 − 3244032, 3244032, 48660480, 48660480 48660480, 48660480 97320960, 97320960, 97320960, 97320960 − 97320960, 97320960, 97320960, 97320960 97320960 64880640, 64880640 194641920, 194641920 194641920 48660480, 48660480, 48660480, 48660480 32440320, 32440320, 32440320, 32440320 97320960, 97320960, 97320960, 97320960 97320960 194641920 97320960, 97320960, 97320960, 97320960 19464192, 19464192 97320960 97320960, 97320960 97320960, 97320960, 32440320, 32440320 97320960, 97320960, 97320960, 97320960 194641920, 194641920 194641920, 194641920 97320960, 97320960, 97320960, 97320960 97320960 64880640, 64880640 194641920, 194641920 194641920, 194641920 97320960, 97320960, 97320960, 97320960 194641920, 194641920 194641920, 194641920 194641920, 194641920 194641920, 194641920 4.5. Monster Sections 4.5 82 Monster Sections In the 1970s Bernd Fischer during his research on a 3-transposition found what are called now the Fischer groups F i22 , F i23 and F i′24 . They are defined by the finite groups G satisfying the following : • G is generated by a G-conjugacy class X with the property that if x1 , x2 ∈ X, then order(x1 x2 ) 6 3; • The derived subgroups G′ and G′′ are equal; • If H 6 G such that H is a 2- or 3-subgroup, then H 6 Z(G). Indeed, Fischer defined the commuting involution graph on X to prove that G′ is simple. Now, According to Table 4.1 our aim is to construct the commuting involution C(G, X) where (G = 2.F i22 , X = 2F ), (G = 2.F i22 .2, X = 2B), (G = 2.F i22 .2, X = 2C) and (G = 2.F i22 .2, X = 2F ). In fact, we are going to apply Algorithm 2 for the cases (G = 2.F i22 .2, X = 2B), (G = 2.F i22 .2, X = 2C), while we will use Algorithm 3 to calculate C(G, X) in the cases (G = 2.F i22 , X = 2F ) and (G = 2.F i22 .2, X = 2F ). • The next table considers the structure of the commuting involution graph for 2.F i22 , with the class 2F . The calculations in this table involve CG (t)-orbit sizes. Table 4.8: C(2. F i22 , 2F ) Class 2A 2BC 2DE 2F 3A 3B 3C 3D 4A ∆1 (t) 1 48 576, 216, 144, 9, 144 432,216,432,432,432,432, 216,6912,216,432,216 − − − − − ∆2 (t) − − − − ∆3 (t) − − − − 1536, 384, 384 8192 12288, 6144, 6144, 12288 49152, 49152 6912,576,6912,576, 9216,576,6912,6912,576,9216 − − − − − Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) Class 4BC ∆1 (t) − 4D 4EF − − 4G − 5A 6A 6B 6CD 6E 6F G 6H 6IJ 6KL − − − − − − − − − 6M N 6OP 6Q 6R − − − − 6S 6T U − − 6V W − 6XY 7A − − 8A − 8B − ∆2 (t) 6912,6912,6912,6912, 6912,6912 6912, 6912, 6912, 6912, 55296 55296,6912,6912,6912,55296, 6912,6912,6912 13824,13824,6912,6912,1152, 6912,13824,1152,13824,1152, 1152,6912,13824,6912,13824, 6912,6912,6912,13824,6912, 6912,13824,13824,6912,13824 ,13824,6912,13824 − 384, 384, 1536 8192 4608, 4608, 4608, 4608 6144, 12288, 6144, 12288 24576 49152, 49152 73728 9216,9216,55296,6912,13824, 9216,9216,6912 36864, 36864, 36864, 36864 221184 110592, 110592 55296,55296,13824,55296, 13824,13824,13824,55296 110592, 110592 36864,55296,36864,36864, 55296,36864 110592,110592,110592, 110592 442368, 442368 442368,442368,147456, 147456 110592,55296,55296,55296, 110592,110592,55296,110592 18432,110592,18432,110592, 110592,110592,18432,18432 ∆3 (t) − − − − 73728, 73728, 110592, 110592 − − − − − − − − − − − − − − − − − − − 83 4.5. Monster Sections 84 Class 8C 8DE ∆1 (t) − − 9A 9B − − ∆2 (t) − 110592,110592,110592, 110592,110592,110592, 110592,110592,110592, 110592,110592,110592 − − 9C 10A − − − − 10BCDE − − 12AB 12C 12D − − − 12E − 12F − 12GHIJKLM N 12O − − 12P 12QR 13AB 14A − − − − 55296, 55296, 55296, 55296 73728, 73728, 73728, 73728 110592,55296,55296,55296,110592, 110592,110592,55296 55296,55296,55296,55296, 55296,55296,55296,55296 110592,110592,55296,55296,110592, 55296,110592,55296 221184, 221184, 221184, 221184 110592,110592,110592, 110592,110592,110592, 110592,110592,110592, 110592,110592,110592 221184,221184,221184, 221184 442368,442368,442368, 442368 − − 14BC − − 15A − − 18A 18B − − − − ∆3 (t) 221184, 221184 − 147456, 147456 147456,147456, 442368, 442368 884736, 884736 73728,110592, 110592, 73728 221184,221184, 221184,221184 221184 − − − − − − − − 884736, 884736 147456,442368, 442368,147456 442368,442368, 442368,442368 442368,442368, 442368,442368 147456, 147456 442368,442368, 147456,147456 Chapter 4. Class 18GH 18IJ 18K 20AB 21A 24AB 26AB 30ABC 42A Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) ∆1 (t) − − − − − − − − − ∆2 (t) − − − − − − − − − 85 ∆3 (t) 442368, 442368, 442368, 442368 442368, 442368 884736, 884736 442368, 442368, 442368, 442368 884736, 884736 442368, 442368, 442368, 442368 884736, 884736 442368, 442368, 442368, 442368 884736, 884736 • In the final three tables we study the commuting involution graphs for the group 2.F i22 .2 , with the classes 2B, 2C and 2F . We end up with discs sizes. Table 4.9: C(2. F i22 .2, 2B) Class 2A 2DE 3A 6A ∆1 (t) 1 693 − − ∆2 (t) − − 2816 2816 Table 4.10: C(2. F i22 .2, 2C) Class 2A 2DE 3C 4D 6C ∆1 (t) 1 1575 − − − ∆2 (t) − − 22400 75600 22400 Table 4.11: C(2. F i22 .2, 2F ) Class 2A 2DE 2G 3A 3B 3C ∆1 (t) 1 27, 1080, 540 3240, 3240 − − − ∆2 (t) − − − 2304 − 11520, 5760 ∆3 (t) − − − − 5120 − 4.5. Monster Sections 86 Table 4.12: C(2. F i22 .2, 2F ) Class 4D 4F 4G 4I 4K 5A 6A 6B 6C 6JK 6LM 6T U 6V W 7A 8A 8E 8F ∆1 (t) − − − − − − − − − − − − − − − − − ∆2 (t) 17280, 17280, 2160, 2160, 6480 103680, 51840 51840, 51840 51840, 51840 103680, 103680, 17280, 17280 414720, 27648 2304 − 5760, 11520 − 34560, 34560, 103680 103680 69120, 69120, 103680, 103680 − 207360, 207360 207360, 207360, 69120, 69120 − 9AB 10A 10CD 11A 12EF 12G − − − − − − − 414720, 27648 414720, 414720, 414720, 414720 − 414720 − 12LO 12T 12U 14A 15A 16B − − − − − − − 414720, 414720, 414720, 414720 − − − − 18AB 18GH 22B 30A − − − − − − − − ∆3 (t) − − − − − − − 5120 − 138240 − − 138240 552960 − − 414720, 414720, 829440 552960 − − 3317760 − 138240, 138240, 138240, 138240 414720, 414720 − 829440, 829440 552960 1658880 1658880,1658880, 1658880,1658880 552960 1658880 3317760 1658880 Chapter 4. Commuting Involution Graphs of C(2. G, X) and C(2. G.2, X) 87 The next theorem gives the structure of the commuting involution graph C(G, X), where (G, X) is described in Table 4.1 Theorem 4.5.1. For G one of the groups of Table 4.1, we have the following results: • DiamC(2.M12 .2, 2D) = 5 and |∆1 | = 31, |∆2 | = 360, |∆3 | = 960, |∆4 | = 230, |∆5 | = 2. • DiamC(2.M22 .2, 2F ) = 3 and |∆1 | = 51, |∆2 | = 800, |∆3 | = 1920. • DiamC(2.Suz.2, 2D) = 4 and |∆1 | = 991, |∆2 | = 229680, |∆3 | = 3950100, |∆4 | = 536668. • DiamC(2.HS.2, 2B) = 3 and |∆1 | = 211, |∆2 | = 1932, |∆3 | = 56. • DiamC(2.HS.2, 2E) = 4 and |∆1 | = 131, |∆2 | = 5660, |∆3 | = 37240, |∆4 | = 3168. • DiamC(2.Co1 , 2D) = 3 and |∆1 | = 138799, |∆2 | = 2521524544, |∆3 | = 18839494656. • DiamC(2.F i22 , 2F ) = 3 and |∆1 | = 12643, |∆2 | = 26511616, |∆3 | = 46448640. • DiamC(2.F i22 .2, 2B) = 2 and |∆1 | = 1387, |∆2 | = 5632. • DiamC(2.F i22 .2, 2C) = 2 and |∆1 | = 3151, |∆2 | = 120400. • DiamC(2.F i22 .2, 2F ) = 3 and |∆1 | = 9775, |∆2 | = 9591984, |∆3 | = 29317120. Proof. The proof of this theorem follows from Lemma 4.2.1 and the previous tables. Chapter 5 Finite Groups Of Lie-type In this chapter we give a summary about the finite groups of Lie-type, including the construction and different widely known properties of such groups. This will put us in good position for the next chapters, where we are going to investigate the commuting involution graph of some of those groups. Finite groups of Lie-type such as the classical matrix groups and their projective group may be defined in a variety of ways. Nevertheless, to completely realise the structure of these groups, it is beneficial to show them in the scope of comprehensive theory. The method that we are going to consider is to look at such groups algebraically. 5.1 Algebraic Group Theory Now we will give a summary of some preliminary results related to algebraic groups. As our purpose will be to quickly familiarize ourselves with those results we may utilize in the coming chapters, we shall often full details of proofs and definitions may not be provided. For the most accurate understanding of these topics we point to [43, 58]. Assume that k is an algebraically closed field. A subset V ⊆ k n ( the vector space on n-tuples over k ) which is defined as the set of points which vanish on a finite set of polynomial equations with coefficients in k is called an affine variety over k. The group G is called an affine algebraic group if G is an affine variety and the maps 88 Chapter 5. Finite Groups Of Lie-type 89 G × G −→ G , (x, y) 7−→ xy; and G −→ G , x 7−→ x−1 are morphisms of affine varieties. Furthermore, the Zariski topology on V is a topology space with closed sets defined to be the subvarieties of V . We should note that in sections 5.1-5.4 we take k to be an algebraically closed field. The typical example for G is the special linear group, SLn (k) which is defined as 2 SLn (k) = {(aij ) ∈ k n : det(aij ) − 1 = 0}. Any closed subgroup of the general linear group GLn (k) is an affine algebraic group, known as a linear algebraic group. As a matter of fact, it can be demonstrated that every affine algebraic group is isomorphic to a linear algebraic group (and vice versa) see [58], therefore we shall drop the term “algebraic group” to describe a linear or affine algebraic group. Thus one can use matrices properties when looking at elements of algebraic groups. 5.2 Subgroups of Algebraic Group Let x ∈ End(V ) where V is a finite dimensional vector space over an algebraically closed field k. Then x is semisimple if it is diagonalisable, and unipotent if the matrix x has value 1 for all of its eigenvalues. The Jordan decomposition of these elements are essential, and works as follows: Theorem 5.2.1. [28] For x ∈ End(V ), x may be uniquely written as x = xs xu , such that xs is semisimple, xu is unipotent and [xu , xs ] = Id(End(V )) . Proof. See 1.4, [28] Since an algebraic group G is a subgroup of GL(V ) for some V , so that it makes sense to study the semisimple and it is unipotent elements of G. The unique maximal 5.3. Groups with a BN -pair 90 closed connected soluble normal subgroup of an algebraic group G is called the radical of G and denoted by R(G). Furthermore, the unipotet radical of G, Ru (G) define to be the unique maximal closed connected normal unipotent subgroup of an algebraic group G. Moreover, G is reductive if Ru (G) = 1 and semisimple if R(G) = 1. Also the algebraic group G is said to be simple if it has no proper closed connected normal subgroups. Let k ∗ be the multiplicative group of k and let T be a subgroup of G such that T ∼ = k ∗ × · · · × k ∗ . Then T is said to be a torus of G. The elements of T are semisimple. In fact, for x a semisimple element of G there is a torus T of G containing x. A maximal closed connected soluble subgroup of G is known as a Borel subgroup of G. The next theorem establishes some principal results related to tori and Borel subgroups of G: Theorem 5.2.2. Let G be a linear algebraic group defined over an algebraically closed field. The following hold: 1- All maximal tori in G are G-conjugate; 2- Any maximal torus lies in some Borel subgroup of G; 3- Any two Borel subgroups of G are G-conjugate; 4- If G is connected and B is a Borel subgroup of G then NG (B) = B. Proof. See [43] 5.3 Groups with a BN -pair Tits [66] presented the notion of a group with a BN -pair, and it has a vital role to play in understanding the algebraic groups. Now we establish the main definition, followed by the related results concerning algebraic groups. Definition 5.3.1. A pair of subgroups B and N of a group G are called a BN -pair if the following axioms hold: 1. B and N generate G; Chapter 5. Finite Groups Of Lie-type 91 2. If K = B ∩ N , then K ⊳ N ; 3. The group W = N/K generate by a finite set of involutions S; 4. For ns ∈ N maps canonically to 1 6= s ∈ W , ns not normalise B; 5. For any s ∈ S and n ∈ N , we have ns Bn ⊆ Bns nB ∪ BnB. The finite group W is called the Weyl group of G. Moreover, the pair (W, S) is called a Coxeter system. Now if G is a connected reductive linear algebra group. Associated with W is a root system Φ = Φ+ ∪ Φ− with positive roots Φ+ and negative roots Φ− . Moreover, corresponding to the root system Φ is a system of fundamental root {α1 , α2 , . . . αn } which leads to the definition of the Dynkin diagram of G. For each root α ∈ Φ there corresponds a subgroup (called a ”root subgroup” ) Xα of G. For further details in that regard we refer the reader to ([28], 1.11). On the other hand, a BN -pair of an algebraic group G is said to be a split BN -pair if it satisfies the following conditions: • B and N are closed subgroups of G; • B = U (B ∩ N ) is a semidirect product of a closed normal unipotent group U and a closed commutative subgroup B ∩ N , all of whose elements are semisimple; • B∩N = T n∈N nBn−1 . One can prove that a connected reductive group G over algebraic closed field contains a split BN -pair (see [43], for instance). Now suppose that B is a Borel subgroup of G and T is a maximal torus of G such that T ⊆ B. Thus if U = Ru (B), then B = U T and B ∩ N = T . For clarity, suppose that G = SLn (k). Then if we let B to be the upper triangular matrices, then B is Borel subgroup of G and the group of monomial matrices is N . The subgroup of diagonal matrices is T and the upper uni-triangular matrices form U . 5.4. Classification of Simple Algebraic Groups 5.4 92 Classification of Simple Algebraic Groups The simple algebraic groups over an algebraically closed field k are classified by their corresponding connected Dynkin diagram. There is an associated Dynkin diagram to each connected reductive algebraic group G. If a Dynkin diagram of a connected semisimple algebraic group is connected then G must be simple and of one of the types presented in Figure 5.1. The group G has a unique Dynkin diagram but the converse is not ture. For instance, the groups P GLn+1 (k) and SLn+1 (k) have the same Dynkin diagram of type An . Full information regarding the Dynkin diagram may be found in [28]. Al , l > 1 Bl , l > 2 Cl , l > 3 Dl , l > 4 E6 E7 E8 F4 G2 ◦ − ◦ −··· − ◦ − ◦ α1 α2 αl−1 αl ◦ − ◦ −··· − ◦ ⇒ ◦ α1 α2 αl−1 αl ◦ − ◦ −··· − ◦ ⇐ ◦ α1 α2 αl−1 αl ◦α l | ◦ − ◦ −··· − ◦ − ◦ α1 αl−2 αl−1 α2 ◦α6 | ◦ − ◦ − ◦ − ◦ −◦ α1 α2 α3 α4 α5 ◦α7 | ◦ − ◦ − ◦ − ◦ − ◦ −◦ α1 α2 α3 α4 α5 α6 ◦α 8 | ◦ − ◦ − ◦ − ◦ − ◦ − ◦ −◦ α1 α2 α3 α4 α5 α6 α7 ◦ −◦ ⇒ ◦ − ◦ α1 α2 α3 α4 ◦⇛◦ α1 α2 Figure 5.1 Connected Dynkin diagrams Chapter 5. Finite Groups Of Lie-type 5.5 93 Finite Groups of Lie-type The previous sections were associated with algebraic groups over an algebraically closed field. The finite group of Lie-type defined as a subgroups of connected reductive algebraic groups over an algebraic closed field of prime characteristic. This section will be concerned with some basic concepts associated with finite groups of Lie-type. we start with the abstract definition and then study some properties related to these groups. Let k be the algebraic closure of the finite field Fp , where p > 0 is a prime. Let G is a connected reductive group over k. Thus, G is isomorphic to closed connected subgroup of GLn (k) for some n. Set q = pr for some r > 1 and define the map Fq : GLn (k) −→ GLn (k) as follows Fq : (aij ) 7−→ (aqij ). Now, a homomorphism F : G −→ G, is called a standard Frobenius map if there exist an injective homomorphism map i : G −→ GLn (k) for some n, satisfying i(F (g)) = Fq (i(g)) for all g ∈ G and some q = pr . Generally, A Frobenius map is a homomorphism F : G −→ G, with the condition that F j , j > 1, is a standard Frobenius map. Moreover, if F is a standard Frobenius map the fixed point group GF = {g ∈ G : F (g) = g}. is finite. If G is a connected reductive algebraic group and F is a Frobenius map the finite group GF which arise in this way is called a finite group of Lie-type. A classification of finite, simple groups of Lie-type can be obtained from the classification of simple algebraic groups. Nonetheless in some situations, the graph automorphism of Dynkin diagrams together with an appropriate field automorphism yields extra simple groups, when the field is finite. In the next chapter a complete list of finite simple groups of Lie-type will be included. 5.5. Finite Groups of Lie-type 94 Now, suppose that G be a connected group over an algebraically closed field k of characteristic p, and let F : G −→ G is an epimorphism such that GF is finite. Then the map δ : G −→ G define as follows δ(x) = x−1 F (x) for all x ∈ G. is surjective. This result known as the Lang-Steinberg theorem has a vital role in the theory of finite group of Lie-type see [28]. Let G be a connected reductive group with a Frobenius map F : G −→ G. Then we aim to prove that G possess an F -stabilizer Borel subgroup. To prove that we take any Borel subgroup B of G, as any two Borel subgroup are conjugate in G so that B g = gBg −1 for some g ∈ G. Also, F (B g ) = B g if and only if F (B)g −1 F (g) = B. But F (B) is also a Borel subgroup of G, thus there is h ∈ G such that F (B)h = B. Now by applying Lang-Steinberg theorem we obtain h = g −1 F (g) for some g ∈ G. Therefore B g is an F -stabilizer Borel subgroup of G. A Borel subgroup of GF is defined to be B F , where B is an F -stabilizer Borel subgroup of G. One can prove that in a similar way that any F -stabilizer Borel subgroup contains an F -stabilizer maximal torus. A maximal tours of GF is defined to be a subgroup of the form T F , such that T is a F -stabilizer maximal tours of G lies in an F -stabilizer Borel subgroup B F , for such a torus T F , we get N = NG (T F ) is stable under F . Consequently, we obtain a split BN -pair for GF from B F and N F . Furthermore, it is not necessarily true that an F -stabilizer maximal torus of G lies inside an F -stable Borel subgroup of G, so that not every maximal torus of GF lies inside a Borel subgroup of GF . A maximally split is an F -stabilizer maximal torus of G which lie in an F -stable Borel subgroup of G, whereas a torus of the form T F , such that T is a maximal split torus of G is called maximal split torus of GF . Chapter 5. Finite Groups Of Lie-type 95 Now we give an important results related to the Borel subgroup B F and maximal torus of the group of Lie-type GF : Theorem 5.5.1. For a finite group of Lie-type GF , the following properties hold: 1. Any two Borel subgroups of GF are GF -conjugate; 2. Any two maximal split torus subgroups of GF are GF -conjugate; 3. Any two maximal tori which lie in B F are B F -conjugate. Proof. [43]. Chapter 6 Commuting Involution Graphs of Exceptional Groups of Lie-type 6.1 Introduction In the previous chapter, preliminary results relating to finite groups of lie-type were displayed in abbreviated form. Our target in this chapter is to study the commuting involution graphs, C(G, X), of a particular type of exceptional group of lie-type. One can see from the classification of finite simple groups (see [31, 67] ) that exceptional groups of Lie-type divide into two types: • Untwisted groups: (An (q), n > 1), (Bn (q), n > 2), (Cn (q), n > 3), (Dn (q), n > 4), E6 (q), E7 (q), E8 (q), F4 (q), G2 (q). • Twisted groups: (2 An (q), n > 1), (2 Dn (q), n > 1), 3 D4 (q), 2 E6 (q), (2 B2 (22n+1 ) = Sz(2n+1 ), n > 1), (2 G2 (32n+1 ) = Ree(32n+1 ), n > 1), (2 F4 (22n+1 ), n > 1), 2 F4 (2)′ . Our investigation involves analyzing the disc structure and determining the diameters of the commuting involution graphs, C(G, X), when G is one of the following groups: • Untwisted groups : G2 (5), F4 (2), E6 (2). • Twisted groups : R(27), 2 F4 (2)′ , 3 D4 (2), 3 D4 (3). 96 Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type 97 Let G is one of the groups above. Let t ∈ X where X is a conjugacy class of involution. To calculate the disc structure and the diameter for the commuting involution graph C(G, X) we may use an approach similer to the one implemented in Chapter 3. However, we may need in some cases to apply a new algorithm to determine C(G, X). The permutation character of CG (t) in G has been found using the Magma code “PermutationCharacter” and hence the number of CG (t)-orbits (permutation rank on X) under the action of X on CG (t) is calculated. Moreover, the complex character tables of G are available in the GAP library. Therefore, we may use them to determine the size of X and then use the Magma code “CentralizerOfInvolution” to compute CG (t). For our aforementioned groups the sizes of conjugacy classes of involutions and the permutation ranks on each class of involution are listed in the next table. Table 6.1: Disc sizes and Permutation Character Group G2 (5) F4 (2) F4 (2) F4 (2) F4 (2) E6 (2) E6 (2) E6 (2) R(27) 2 F (2)′ 4 2 F (2)′ 4 3 D (2) 4 3 D (2) 4 3 D (3) 4 Class 2A 2A 2B 2C 2D 2A 2B 2C 2A 2A 2B 2A 2B 2A Size of Class 406875 69615 69615 4385745 350859600 5081895 8822169720 1587990549600 512487 1755 11700 819 68796 43584723 Permutation Rank 69 5 5 33 1002 5 62 719 54 5 30 4 27 209 6.2. Algorithms 98 One can note from Table 6.1 that many difficulties may arise during our study of C(G, X) when G is F4 (2) and X = 2D, and when G is E6 (2) and X = 2C. The reason for this is that these groups have large permutation rank on X, being respectively equal to 1002 and 719 with big class size (see [67]). Consequently, we shall initialize a new algorithm to determine the suborbits. However, Algorithm 2,3 which we established in Chapter 4, will be employed to find the CG (t)-orbit on X for the remaining conjugacy classes of involutions. 6.2 Algorithms Let x ∈ X such that CCG (t) (x) ∩ X = ∅. By Theorem 4.2.2 part 4 we have that d(x, t) > 3. In order to show that x ∈ ∆3 (t), it suffices to find an element y ∈ CG (x) such that ty ∈ C, for some subset XC of ∆2 (t). Additionally, the Magma code “CentralizerOfInvolutuion” will be used to compute the centralizer of the involutions in G and the code “RandomElementOfOrder” will be employed to find a random element of given order. We apply the ideas above in the following algorithm to find CG (t)-orbits in ∆3 (t): Algorithm 4 . Input: G is either E6 (2) or F4 (2), t an involution in 2C or 2D respectively; i: set Rep(∆2 (t)) −→ the set of representative for C such that XC ⊆ ∆2 (t). ii: for x 7−→ Random(X) do if CCG (t) (x) ∩ X = ∅ then iii: repeat CInvolution −→ CentralizerOf Involutuion(G, x), in case G = E6 (2) and CInvolution −→ Centralizer(G, x), for G = F4 (2). iv: stop when the size of CInvolution equal to CG (t). v: repeat y −→ RandomElementOf Order(CInvolution, 2) Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type vi: stop when #F ix(y) equal to 256 in case G = F4 (2) 99 or Dimension(Eigenspace(y, 1)) equal 15 in case G = E6 (2). vii: if order(t ∗ y) equal to order(z) for some z in Rep(∆2 (t)) then x ∈ ∆3 (t). Output: the set of suborbit representatives in ∆3 (t) . 6.3 CG (t)-orbits Untwisted Groups Our aim is to calculate the commuting involution graphs for the untwisted groups G2 (5), F4 (2) and E6 (2). The complex character tables for these groups are available, therefore information about the number of conjugacy classes of involutions and the size of XC can be acquired. For the groups G2 (5), F4 (2) we are going to use the permutation representations with 3906 and 6988 points respectively from The Online Atlas. Meanwhile, we consider E6 (2) as a matrix representation of dimension 27 over GF (2). Moreover, from Table 6.1 we see that there is only one class of involutions namely 2A in G2 (5). By calculation, for t ∈ 2A the size of the fixed-point set is 42. We also see that F4 (2) contains four classes of involution, namely 2A, 2B, 2C and 2D with size of fixed-point set equal to 5376 in 2A and 2B and 1280, 256 in 2C, 2D respectively. Additionally, E6 (2) has three classes of involutions with dimension of fixed spaces being respectively equal to 21, 17, 15 (information about fixed spaces will be available in the next chapter as we use it to make a suborbit invariant). For a deeper understanding of these groups we refer the reader to [27, 28, 71]. Now we provide tables giving the full details about the CG (t)-orbits for t an involution in G2 (5), F4 (2) or E6 (2). Furthermore, we include the sizes of the suborbits in each ∆i (t) belongs to. 6.3. CG (t)-orbits 100 Table 6.2: C(G2 (5), 2A) Class 2A 3A 3B 4AB 5A 5B 5C 5D 5E 6AB 6C 7A 8AB 10AB 10CD 12AB 15A 15B 15E 20AB 21AB 24ABCD 25A 30AB 31ABCDE ∆1 (t) 200, 450 − − − − − − − − − − − − − − − − − − − − − − − − ∆2 (t) − 120 1200, 1800 1200, 1800 144 720 480, 1440 − 1440, 1440 1200, 1800 3600, 3600, 3600, 3600 − 7200, 7200 3600 7200, 7200 7200, 7200 1440, 1440 − 7200, 7200 7200, 7200 − 7200, 7200 − 7200, 7200 − Table 6.3: C(F4 (2), 2A) Class 2A 2C 3A 4C ∆1 (t) 270 2016 ∆2 (t) 32768 34560 ∆3 (t) − − − − − − − 960 − − − 14400 − − − − − 14400 − − 14400 − 14400 − 14400 Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type Table 6.4: C(F4 (2), 2B) Class 2A 2C 3A 4D ∆1 (t) 270 2016 ∆2 (t) 32768 34560 Table 6.5: C(F4 (2), 2C) Class 2AB 2C 2D 3AB 4AB 4CD 4F 4JK 4L 4M 5A 6GH ∆1 (t) 30 180,1920,1920,32,32 720,960,720,960,960,960,11520 ∆2 (t) 32768 15360 11520 1024,1024 30720,30720 737280 184320,184320 1048576 983040 Table 6.6: C(F4 (2), 2D) Class ∆1 (t) 2AB 192, 72, 3, 72, 12 2C 144,24,144,144,576,192,144,72,576,24,144,9,72 ,12,144,576,144,72,192,72,576,12 2D 1152,144,576,576,144,144,576,144,144,1152, 1152,576,576,144,24,576,144,1152,24,576,144, 576,144,1152,1152,144,144,1152,144,144,1152, 576,144,576,144,576,576,576,144,144,576,576, 576,144,144,9216,144,576,1152,144,144,1152, 1152,24,144,24,1152,144,144,576,576,576,576, 1152,576,1152,576,144,1152,144,144,576,144, 1152 ∆2 (t) ∆3 (t) 101 6.3. CG (t)-orbits Class 3AB 3C 4AB 4CD 4EF 4GH 4I 4JK 4L 4M ∆1 (t) 102 ∆2 (t) 2048,24576,6144 262144 576,12288,1152,576,9216,576,1152, 576,576,576,1152,1152,576,192,576 576,1152,1152,144,192,576,2304,576, 576,4608,576,144,4608,2304,576,288,4608,576, 576,2304,12288,576,288,192,288,576,576,288, 144,576,4608,192,2304,144,576 9216,9216,576,2304,1536,576,576,9216,1536, 9216,2304,576,9216,9216,1536,9216,1536,9216 ,2304,2304 2304,4608,2304,2304,2304,4608,18432,4608, 9216,9216,4608,18432,18432,73728,4608, 18432,4608 18432,9216,18432,73728,9216,18432,9216,9216 ,36864,18432,18432,9216,9216,18432,9216, 9216,73728,18432,9216,9216,9216,9216,36864 ,9216,36864,36864,9216,18432 18432,9216,18432,4608,18432,4608,1152, 18432,18432,18432,4608,4608,18432,9216, 18432,18432,9216,18432,4608,4608,18432, 9216,9216,9216,2304,9216,18432,1536, 18432,18432,4608,18432,9216,1152,1152, 9216,4608,18432,1152,18432,18432,4608, 4608,4608,9216,9216,1536,4608,9216,4608, 1536,4608,9216,4608,9216,4608,2304,4608, 4608,18432,18432,1536,9216,18432,9216, 2304,18432,4608,2304,4608 36864,9216,9216,36864,36864,9216,36864, 9216,36864,9216,147456,9216,9216,9216, 36864,147456,36864,9216,36864 9216,18432,4608,9216,18432,18432,18432,2304 ,4608,4608,18432,9216,36864,18432,2304,9216 ,4608,9216,18432,9216,9216,9216,18432,18432 ,9216,9216,9216,18432,18432,9216,18432,9216, 4608,9216,9216,4608,9216,18432,4608,18432, 9216,18432,9216,18432,18432,9216,18432,9216 ∆3 (t) Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type Class 4M 4N 4O ∆1 (t) ∆2 (t) 18432,18432,4608,4608,18432,4608,18432, 18432,9216,18432,18432,4608,4608,9216, 18432,9216,9216,18432,18432,9216,18432, 9216,18432,18432,9216,36864,9216,18432, 9216,9216,18432,18432,18432,18432 147456, 147456, 147456, 147456 36864,36864,36864,36864,147456,36864,36864, 36864,147456,147456,36864,147456,36864, 36864,36864,36864 5A 6AB 6CD 6EF 6GH 6IJ 6K 7AB 8A 8B 8CDEF 8G 8HI 8J 103 ∆3 (t) 196608, 196608, 589824 6144,24576,73728,73728,73728,6144,24576 36864,294912,73728,49152,36864,49152,73728 ,73728 786432 294912,73728,12288,49152,73728,73728, 147456,49152,147456,73728,36864,73728, 73728,73728,36864 73728,147456,73728,147456,73728,73728, 294912,73728,73728,73728,294912, 73728,294912,147456,147456,294912 2359296 1572864 294912,294912,294912,294912 294912,147456,147456,147456,147456,147456, 294912,294912,147456,147456,294912,147456 73728,73728,73728,73728,73728,294912,73728, 294912,147456,147456,147456,147456,73728, 73728,294912,73728,24576,73728,24576,294912 589824, 589824 294912,294912,294912,294912,294912,294912 294912,294912 589824,589824,589824,589824,589824,589824, 589824,589824,589824,589824,589824,589824, 589824,589824,589824,589824 6.3. CG (t)-orbits Class 8K 104 ∆1 (t) ∆2 (t) 589824,589824,589824,589824, 589824,589824 9AB 10AB 10C 12AB 12CD 12EF GH 12IJ 12KL 12M N 12O 13A 14AB 15AB 16AB 17AB 18AB 20AB 21AB 24ABCD 28AB 30AB 294912,294912,294912,294912 786432,786432 589824,294912,589824,589824, 98304,294912,294912,98304, 589824,294912 294912,294912,294912,294912, 294912,294912,294912,294912 294912,294912,294912,294912 ∆3 (t) 4718592,1572864 589824,589824,1179648,1179648 589824,589824,1179648,1179648, 1179648,1179648 1179648 1179648 2359296,2359296 589824,589824,589824,589824, 589824,589824,589824,589824 589824,589824,589824,589824, 589824,589824 2359296,2359296,2359296,2359296 9437184 4718592 4718592,1572864 2359296,2359296,2359296,2359296 9437184 4718592,4718592 2359296,2359296,2359296,2359296 9437184 2359296,2359296,2359296,2359296 4718592,4718592 4718592,4718592 Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type 105 Table 6.7: C(E6 (2), 2A) Class 2A 2B 3A 4B ∆1 (t) 2790 124992 − − ∆2 (t) − − 2097152 2856960 Table 6.8: C(E6 (2), 2B) Class 2A 2B 2C 3A 3B 4A 4B 4C ∆1 (t) 63,2160,2160 64512,120960,30720,4320,30240,30720, 30240,56 60480, 60480, 725760, 725760, 967680 − − − − − 4D 4E 4F − − − 4J 4K 5A 6A 6D 6F 8C 12B − − − − − − − − ∆2 (t) − − − 2359296 16777216 774144 967680,2211840,967680,2211840, 725760 3870720,3870720,3870720,1935360, 4423680,1935360,7741440,1935360, 7741440,1935360,4423680,4423680, 3870720,4423680 8847360,7864320,7864320 46448640,46448640 61931520,2064384,2064384, 61931520,61931520,61931520 123863040,123863040 743178240 939524096 70778880,70778880 990904320 1056964608 990904320,990904320 1132462080,1132462080 6.3. CG (t)-orbits 106 The final case is C(E6 (2), 2C). Here we shall make a different table for each ∆i (t), because the size of the set XC is too large in most instances. Table 6.9: ∆1 (t) Structure C(E6 (2), 2C) Class 2A 2B 2C ∆1 (t) 3, 84, 1536, 2016 16128, 86016, 43008, 224, 5376, 168, 2016, 8064, 8064, 10752, 32256, 32256, 10752 258048,86016,64512,86016,16128,64512,64512,258048,36864,1032192,36864,36864, 32256,32256,64512,129024,16128,96,86016,129024,16128,32256,36864,32256, 258048,86016,129024,96,5376 Table 6.10: ∆2 (t) Structure C(E6 (2), 2C) Class 3A 3B 3C ∆2 (t) 1572864, 917504 29360128 134217728 4A 4B 64512, 86016, 1032192, 64512, 32256, 786432, 21504, 36864, 36864, 64512, 36864, 1536 129024,1032192,1032192,64512,43008,258048,64512,129024,64512,32256,1032192, 32256,36864,786432,129024,86016,258048,32256,64512,1536,1536,16128,32256 516096,516096,1032192,516096,516096,2064384,64512,2064384,2064384,1032192, 516096,516096,4128768,129024,516096,516096,516096,2064384,258048,4128768, 129024,1032192,4128768,688128,64512,4128768,1032192,2064384,516096,129024, 1032192,64512,2064384,516096,129024,516096,129024,129024,258048,1032192, 2064384,688128,2064384,64512 4C 4D 2752512,1032192,2752512,11010048,1376256,16515072 4E 2064384,4128768,8257536,1032192,4128768,33030144,2064384,516096, 516096,516096,2064384,4128768,2064384,2064384,2064384,4128768,1032192, 4128768,4128768,258048,4128768,516096,2064384,4128768,4128768,4128768, 516096,4128768,2064384,258048,8257536,258048,2064384,4128768,4128768, 2064384,4128768,1032192,516096,1032192,2064384,1032192,1032192,4128768, 516096,1032192,1032192,4128768,516096,4128768,1032192,258048,4128768, 2064384,1032192,4128768,4128768,516096,4128768,4128768,516096 Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type Class 4F 4G 4H 4I 4J 4K 6A 107 ∆2 (t) 16515072,16515072,16515072,4128768,4128768,2752512,2752512,16515072, 5505024,33030144,33030144,33030144,16515072,5505024,2064384,1032192, 4128768,33030144,2064384,4128768,4128768,33030144,4128768,33030144, 16515072,16515072,4128768,16515072,1032192,4128768 8257536,66060288,16515072,16515072,8257536,33030144,16515072,33030144, 4128768,33030144,66060288,4128768,16515072 37748736,66060288,66060288,37748736 16515072,66060288,66060288,33030144,33030144,66060288,33030144, 33030144,16515072,33030144,66060288,264241152,66060288,66060288, 33030144,66060288,88080384,11010048,11010048 8257536,33030144,66060288,16515072,16515072,4128768,16515072,16515072, 16515072,8257536,8257536,8257536,66060288,66060288,33030144,66060288, 4128768,33030144,8257536,16515072,33030144,1376256,33030144,33030144, 33030144,8257536,4128768,33030144,33030144,66060288,66060288,16515072, 33030144,66060288,33030144,33030144,66060288,8257536,33030144,33030144, 33030144,33030144,33030144,33030144,8257536,2064384,33030144,33030144, 1376256,8257536,8257536,33030144,16515072,33030144,4128768,4128768, 66060288,8257536,66060288,2064384,16515072,4128768,8257536,8257536, 8257536,8257536,16515072,8257536 66060288,16515072,8257536,8257536,33030144,33030144,33030144,33030144, 66060288,66060288,264241152,8257536,16515072,33030144,16515072,16515072, 8257536,16515072,33030144,16515072,4128768,8257536,66060288,33030144, 66060288,33030144,16515072,33030144,66060288,33030144,66060288,33030144, 16515072,16515072,66060288,16515072,16515072,33030144,16515072,16515072, 8257536 44040192,44040192,33030144,33030144,2752512,33030144 6B 6C 6D 402653184 528482304,704643072 66060288,132120576,88080384,1835008,66060288,264241152,132120576, 88080384,132120576,528482304,66060288,66060288,176160768,132120576, 88080384 6E 132120576,88080384,37748736,132120576,88080384,528482304, 66060288,66060288,528482304,37748736 528482304,1056964608,88080384,352321536,704643072 2818572288 4227858432,1056964608,1056964608 8455716864 37748736,1572864,1572864,132120576,33030144,528482304,33030144,88080384, 37748736,132120576,66060288,66060288,88080384,528482304 6F 6G 6H 6I 8A 6.3. CG (t)-orbits Class 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 12D 12E 12F 12G 12H 108 ∆2 (t) 66060288,132120576,37748736,44040192,132120576,88080384,44040192, 66060288,528482304,88080384,37748736,528482304 132120576,264241152,33030144,264241152,264241152,66060288,66060288, 132120576,264241152,264241152,264241152,132120576,88080384, 88080384,132120576,66060288,66060288,264241152,132120576, 132120576,264241152,132120576,66060288,132120576,16515072, 66060288,66060288,264241152,132120576,66060288,264241152, 33030144,132120576,132120576,132120576,16515072 264241152,264241152,264241152,264241152,132120576,264241152, 264241152,264241152,264241152,264241152,264241152,132120576, 264241152,264241152,132120576,264241152,264241152,264241152, 264241152,264241152,264241152,264241152,1056964608,264241152, 132120576,1056964608 528482304,528482304,2113929216,176160768,2113929216,528482304, 528482304,176160768 2113929216,2113929216,2113929216,2113929216,2113929216 1056964608,528482304,1056964608,1056964608,528482304,528482304, 1056964608,264241152,1056964608,1056964608,1056964608,264241152, 264241152,1056964608,264241152,1056964608,528482304,1056964608, 1056964608,1056964608 2113929216,2113929216,2113929216 4227858432,4227858432,4227858432,4227858432,4227858432,4227858432 2113929216,2113929216,4227858432,2113929216,4227858432,4227858432, 1056964608,2113929216,1056964608,4227858432,2113929216,2113929216 402653184,402653184 132120576,528482304,132120576,528482304,528482304,528482304 1056964608,528482304,264241152,1056964608,528482304,264241152, 1056964608,264241152,264241152,1056964608,1056964608,1056964608, 1056964608,1056964608,264241152,528482304,1056964608,264241152, 1056964608,1056964608,1056964608,264241152,1056964608,1056964608, 1056964608,1056964608,264241152,528482304 1409286144,1409286144,2113929216,2113929216,4227858432,2113929216, 2113929216 4227858432,4227858432,4227858432 4227858432 5637144576 4227858432,2113929216,2113929216,2113929216,352321536,2113929216, 4227858432,4227858432,4227858432,352321536,2113929216,2113929216 Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type Class 12I 12J 12K 12L 12M 12P 16A 16C 24A ∆2 (t) 8455716864, 8455716864 4227858432,2113929216,2113929216,1056964608,1056964608, 4227858432,4227858432,4227858432,4227858432,2113929216, 4227858432,2113929216,4227858432,2113929216,2113929216, 4227858432 4227858432,8455716864,8455716864,4227858432,1409286144, 1409286144,8455716864,4227858432,8455716864,4227858432 16911433728 16911433728,16911433728 16911433728,16911433728 8455716864,8455716864,8455716864,8455716864 16911433728,16911433728,16911433728,16911433728 8455716864,8455716864,8455716864,8455716864,8455716864, 8455716864,8455716864,8455716864 Table 6.11: ∆3 (t) Structure C(E6 (2), 2C) Class 5A 7C 7D 9A 9B 10A 10B 12B 12E 12F 13A 14G 14H 15C 15D 17A 17B 18A 18B 20A 20B ∆3 (t) 1409286144,234881024 805306368 3221225472 22548578304 9663676416,3221225472 4227858432,2818572288, 2818572288 16911433728,8455716864, 2818572288,4227858432,4227858432 528482304,528482304 2818572288 8455716864 19327352832 16911433728 9663676416 22548578304 7516192768,22548578304 45097156608 45097156608 9663676416,9663676416 67645734912 16911433728,16911433728 33822867456,33822867456,33822867456,33822867456 109 6.3. CG (t)-orbits 110 Table 6.12: ∆3 (t) Structure C(E6 (2), 2C) Class 21G 21H 24B 24C 24D 28K 28L 30E 30F ∆3 (t) 19327352832 45097156608 16911433728,16911433728,16911433728,16911433728 33822867456,33822867456 33822867456,33822867456 9663676416,9663676416 33822867456 22548578304, 22548578304 67645734912 Twisted Groups The complex character tables for the groups R(27), 2 F4 (2)′ , 3 D4 (2) and 3 D4 (3) exist. Therefore, the number of conjugacy classes of involutions and the size of XC are known, both of which are very useful when we consider the commuting involution graph for such groups. The permutation representation of R(27), 2 F 4(2)′ , 3 D4 (2) and 3 D4 (3) come from (The Online Atlas) and the number of points is equal to 19684, 1600, 819 and 26572 respectively. However, from Table 6.1 we see that there is only one class of involutions namely 2A in R(27). By calculation, for t ∈ 2A the size of fixed-point set is 28. The Tits group 2 F4 (2)′ contains two classes of involution namely 2A and 2B with size of fixed-point set is 0, 48 respectively. Additionally, 3 D4 (3) has one class of involution with size of fixed space equal to 116. There are two classes of involution in 3 D4 (2) with number of fixed spaces respectively equals 19, 27. A good resource associated to these groups can be found in [27, 28, 71]. The next tables describe the CG (t)-orbits for t an involution in R(27), 2 F4 (2)′ , 3 D4 (2) and 3 D4 (3). Furthermore, The number of elements in each class is given. Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type Table 6.13: C(R(27), 2A) Class 2A 3A 7A 9C 13ABCDEF 14ABC 19ABC 26ABCDEF 37ABCDEF ∆1 (t) 351, 351 ∆2 (t) ∆3 (t) 728 4914,4914,4914,4914 19656 9828,9828 4914,4914,4914,4914 19656 9828,9828 19656 Table 6.14: C(2 F4 (4)′ , 2A) Class 2A 2B 4C 5A ∆1 (t) 10 ∆2 (t) 80 640 1024 Table 6.15: C(2 F4 (4)′ , 2B) Class 2A 2B 3A 4A 4B 4C 5A 6A 8CD 12AB 13AB ∆1 (t) 12, 3 48,48,12,12,12 ∆2 (t) ∆3 (t) 256,256 192,192 96,96 192,192,96 768 768,768 384,384 768,768 1536 111 6.3. CG (t)-orbits 112 Table 6.16: C(3 D4 (3), 2A) Class 2A 3A 3B 3C 3D 4A 4B 6A 6B 6CD 7ABC 7D 8AB 9A 9B 12AB 13ABCDEF 13GHIJK 14ABC 21ABC 21DEF 26ABCDEF 28ABCDEF 28GHI 39ABCDEF 39GHIJKL 42ABC 52ABCDEF 56ABCDEF 73ABCDEF GHIJKLM N OP QR 78ABCDEF 84ABCDEF 104ABCDEF GHIJKL ∆1 (t) 2106,4536 ∆2 (t) ∆3 (t) 224 8736 6048,8736 8736,11232 9072,8424 58968,117936 18144 235872 235872, 235872 19656 471744 235872, 235872 157248 471744 235872, 235872 19656 471744 58968,117936 78624, 78624 471744 58968,117936 58968,117936 117936, 117936, 117936, 117936 78624, 78624 471744 235872, 235872 235872, 235872 235872, 235872 471744 235872, 235872 235872, 235872 235872, 235872 Chapter 6. Commuting Involution Graphs of Exceptional Groups of Lie-type Table 6.17: C(3 D4 (2), 2A) Class 2A 3A 4A ∆1 (t) 18 ∆2 (t) ∆3 (t) 512 288 Table 6.18: C(3 D4 (2), 2B) Class 2A 2B 3A 3B 4A 4B 4C 6A 6B 7ABC 7D 8A 8B 9ABC 12A 13ABC 14ABC 18ABC 21ABC 28ABC ∆1 (t) 3,24 24,24,4,24,24,4,24, 4,24,4,24,24,24,4,24, 24, 4,24 ∆2 (t) ∆3 (t) 384 512 24,24,24,24,192,24 24,24,24,24,192,24,24, 24,24,24,24 384,384,384,384,384,384 1536 384,384,384,384,384,384 512 3072 384,384,384,384,384,384 384,384,384,384,384, 384,384,384 512,1536 1536,1536 3072 1536 1536,1536 3072 1536,1536 113 6.4. Disc Structure 6.4 114 Disc Structure The next theorem gives the structure of the commuting involution graph C(G, X), where (G, X) is described in Table 6.1: Theorem 6.4.1. For G one of the groups of Table 6.1, we have the following results: • DiamC(G2 (5), 2A) = 3 and |∆1 | = 650, |∆2 | = 261264, |∆3 | = 144960. • DiamC(F4 (2), 2A) = 2 and |∆1 | = 2286, |∆2 | = 67328. • DiamC(F4 (2), 2B) = 2 and |∆1 | = 2286, |∆2 | = 67328. • DiamC(F4 (2), 2C) = 2 and |∆1 | = 20944, |∆2 | = 4364800. • DiamC(F4 (2), 2D) = 3 and |∆1 | = 50511, |∆2 | = 113896448, |∆3 | = 236912640. • DiamC(E6 (2), 2A) = 2 and |∆1 | = 127782, |∆2 | = 4954112. • DiamC(E6 (2), 2B) = 2 and |∆1 | = 2856311, |∆2 | = 8819313408. • DiamC(E6 (2), 2C) = 3 and |∆1 | = 3384671, |∆2 | = 609992912640, |∆3 | = 977994252288. • DiamC(R(27), 2A) = 3 and |∆1 | = 702, |∆2 | = 314496, |∆3 | = 197288. • DiamC(2 F4 (2)′ , 2A) = 2 and |∆1 | = 10, |∆2 | = 1744. • DiamC(2 F4 (2)′ , 2B) = 3 and |∆1 | = 147, |∆2 | = 7712, |∆3 | = 3840. • DiamC(3 D4 (2), 2A) = 3 and |∆1 | = 18, |∆2 | = 288, |∆3 | = 512. • DiamC(3 D4 (2), 2B) = 3 and |∆1 | = 339, |∆2 | = 11112, |∆3 | = 57344. • DiamC(3 D4 (3), 2A) = 3 and |∆1 | = 6642, |∆2 | = 27381536, |∆3 | = 16196544. Proof. The proof of this theorem follows from Lemma 4.2.1 and the previous tables. Chapter 7 Investigation On Commuting Involution Graphs for the Exceptional Groups of Lie-type 2E (2) and E (2) 6 7 7.1 Introduction In this chapter we will study the commuting involution graphs for the exceptional groups of Lie-type 2 E6(2) and E7(2) over GF (2). The factored order of 2 E6 (2) is 236 .39 .52 .72 .11.13.17.19 and the smallest matrix representation over GF (2) is of dimension 78. The smallest matrix representation over GF (2) for E7 (2) has dimension 56 and the factored order of E7(2) is 263 .311 .52 .73 .11.13.17.19.31.43.73.127. As we see these groups have a large matrix representation dimension, besides large order. As a result, the computational approach will resolve most of the calculations for such groups. However, Magma commands which are used in the previous chapters to calculate the size of suborbits or to distinguish between classes of involution will be unattainable. In order to solve this problem we use the dimension of the fixed space and the size 115 7.2. Basic Definitions and Results 116 of the centralizer inside the maximal normal 2-subgroup of the centralizer in G of the involutions to recognize different conjugacy classes of involutions. Moreover, many of theoretical techniques may be applied to compute the sizes of the suborbits. Valuable information about such groups can be found in [27, 49, 71]. 7.2 Basic Definitions and Results Let G be a finite group and t ∈ G be an involution. Set X = tG , a conjugacy class of involutions. Our aim is to investigate the commuting involution graphs C(G, X) and to analyze their disc structure for a finite simple groups. Now we give the definition of the fix space, followed by a criteria to make a suborbit invariant. Definition 7.2.1. Let G be a matrix group and V be a corresponding G-module over a finite field F of positive characteristic. Then the Fix Space of a random element g ∈ G is defined as follows: F ixg = {v ∈ V |vg = v} It can be shown that F ixg is a subspace of V . It is also equal to the eigenspace of ρ(g) associated to the eigenvalue 1 ∈ F, where ρ : G −→ GLn (F ) is a representation of G. Lemma 7.2.2. Let G be a finite group and suppose that t ∈ X, where Xis a conjugacy class of involutions in G. Then the following holds: i- for x ∈ X we have F ixt ∼ = F ixx . ii- for H ⊳ G and x ∈ X, we have StH (F ixt ) ∼ = StH (F ixx ). iii- for x ∈ X and z ∈ xCG (t) , we have dimension(F ixt ∩ F ixx ) = dimension(F ixt ∩ F ixz ). Proof. Since t is conjugate to x, there is h ∈ G such that th = hx. Now let w ∈ F ixt , so that wh ∈ F ixx . Thus the map Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 117 Ψ : F ixt −→ F ixx ; w 7→ wh is well defined. It is straightforward to check that Ψ is an isomorphism. Moreover, if g ∈ StH (F ixt ) then h−1 gh ∈ StH (F ixx ).Therefore, the bijection Υ : StH (F ixt ) −→ StH (F ixx ); g 7→ h−1 gh is well defined and an isomorphism, so that [i] and [ii] are proved. Additionally, [iii] follows from [i]. Let O2 be the maximal normal 2-subgroup of CG (t). The group O2 , has a substantial role to play in examining whether the elements of X lie in same class under the action of CG (t). The following lemma elucidates this: Lemma 7.2.3. Let G be a finite group and let t ∈ X, where X is a conjugacy class of involutions in G. Suppose that x, y ∈ X, such that xCG (t) = y CG (t) . For any normal subgroup H of G the following holds: i CH (x) ∼ = CH (y). ii Let z ∈ H and Z = z CG (t) . We have |Z ∩ CH (x)| = |Z ∩ CH (y)|. Proof. As xCG (t) = y CG (t) . We have x conjugate to y in CG (t), so there is h ∈ CG (t) such that hxh−1 = y. Thus if w ∈ CH (x) then wh ∈ CH (y). For that reason the map ς : CH (x) −→ CH (y), w 7→ wh is isomorphism, and the prove of [i] is done. Part [ii] follows form [i]. 7.3 The Commuting Involution Graphs of 2 E6 (2) Let G be the 78-dimensional matrix representation over GF (2) of 2 E6 (2). GAP provides us with enough information about the number of conjugacy classes of involutions and the size of XC as the character table of 2 E6 (2) is available. However, no details about the permutation rank are available for this group. Moreover, the Magma code 7.3. The Commuting Involution Graphs of 2 E6 (2) 118 “IsConjugate” does not work. Therefore, alternative ways to deal with these difficulties will be established based on Lemma 7.2.2 and Lemma 7.2.3. The exceptional group 2 E6 (2) has three classes of involution namely 2A, 2B and 2C with sizes respectively 3968055, 3142699560, and 1319933815200. The dimension of the fix space of these classes is 56, 46 and 40 respectively. Here the study of the commuting involution graph will be limited to the classes 2A and 2B. However, we shall show that C(2 E6 (2), 2C) is connected. The full details about the C(2 E6 (2), 2A) and C(2 E6 (2), 2B) will be given in the next tables, including CG(t)-orbits and discs structure. Table 7.1: C(2 E6 (2), 2A) Class 2A 2B 3A ∆1 (t) 1782 44352 − ∆2 (t) 4A − 1824768 2097152 In contrast to C(2 E6 (2), 2A), the commuting involution graph C(2 E6 (2), 2B) is very complicated. We apply Lemma 7.2.2 and Lemma 7.2.3 to distinguish between the CG(t)-orbits. The method adopted for t ∈ 2B involves the following steps: i- Figure out ∆1 (t) by working inside CG (t) and then using Theorem 4.2.2 part 2. ii- Find the dimension of fix space of the CG (t)-orbits. iii- Calculate O2 = pCore(CG (t), 2) and for z ∈ 2B find CO2 (z). iv- Compute the CG (t)-classes in O2 ∩ ∆1 (t). v- Check the position of the CG (t)-orbits by Algorithm 3 and Algorithm 6 (which will be incorporated later in this chapter). From step[iv] we see there are 6-classes of in O2 ∩ ∆1 (t), distributed as follows; one class in X2A with size 63, in X2B there are 4-classes with size respectively 72, 4320, Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 119 4320 and 4320. Finally in X2C there is just one class with size 60480. Now for xi C a representative of this class and i = 1, 2, ..., 6 , let Ni = |xi G(t) ∩ CO2 (tCG (t) )|. The following table gives the analyzes of C(2 E6 (2), 2B). For x ∈ X we give information about the orbit xCG (t) including the size of the orbit and its position in the ∆i (t). Details are given in the table below: Table 7.2: C(2 E6 (2), 2B) Class 2A 2B 2B 2B 2B 2B 2B 2C 2C 2C 3A 3B 4A 4B 4D 4E 4F 4H 4H 4L 4M 4M 5A 6C 6E |F ix(t∗x) | 56 46 46 46 46 46 46 40 40 40 30 30 36 36 30 30 30 26 26 20 24 24 18 20 18 ∆1 (t) 63 72 40320 4320 4320 4320 64512 60480 967680 60480 ∆2 (t) 1835008 16777216 241920 1290240 8847360 8847360 8847360 2064384 2064384 247726080 123863040 123863040 1207959552 330301440 1056964608 |CO2 (x)| 16777216 16777216 262144 524288 524288 524288 16384 524288 16384 262144 1 1 262144 16384 256 256 256 16384 512 128 128 256 1 128 1 N1 63 63 15 63 63 63 31 63 15 15 0 0 20 12 0 0 0 16 0 0 0 0 0 0 0 N2 72 72 24 72 72 72 0 72 16 24 0 0 20 20 0 0 0 16 0 0 0 0 0 0 0 N3 4320 4320 672 2272 480 480 240 480 240 672 0 0 240 240 8 8 64 0 30 6 14 4 0 0 0 N4 4320 4320 672 480 2272 480 240 480 240 672 0 0 240 240 64 8 8 0 30 6 14 4 0 0 0 N5 4320 4320 672 480 480 2272 240 480 240 672 0 0 240 240 8 64 8 0 30 6 14 4 0 0 0 The diameter of the commuting involution graphs and the disc structure is shown in the following next theorem. The above tables prove this theorem. Theorem 7.3.1. Assume that G is the exceptional group of Lie-type 2 E6 (2) and X is the conjugacy class of involution 2A or 2B in G. The discs structure of C(G, X) is as N6 60480 60480 4224 6720 6720 6720 1440 8512 1440 4224 0 0 5520 1440 56 56 56 2160 180 60 36 124 0 78 0 7.3. The Commuting Involution Graphs of 2 E6 (2) 120 follows: • if X = 2A then DiamC(G, X) = 2 and |∆1 | = 46134, |∆2 | = 3921920. • if X = 2B then DiamC(G, X) = 2 and |∆1 | = 1206567, |∆2 | = 3141492992. Finally, we will prove the connectivity of the C(2 E6 (2), 2C) as follows: The online Atlas provides the 78-dimensional matrix representation of the group G = 2 E6 (2) such that G = hx, yi where x has order 2 and y of order 3. Using a random search for an involution t ∈ G whose fix space has dimension 40. Thus t ∈ 2C. Moreover, tx ∈ ∆1 (t) and by apply Algorithm 6 (which will be incorporated later in this chapter) we get ty ∈ ∆2 (t). Now for a ∈ CC there a path between t and a so that if we conjugate this path by either x or y we get a new path between t and ax and the other between t and ay . Consequently, x, y ∈ SG (CC ) and by Lemma 3.5.1 we obtain C(2 E6 (2), 2C) is connected. Chapter 7. 7.4 Commuting Involution Graph of 2 E6 (2) and E7 (2) 121 The Commuting Involution Graphs of E7 (2) Assume that G is the 56-dimensional matrix representation over GF (2) of E7 (2). Let t ∈ G be an involution and X = tG . As we have seen in Lemma 4.2.6 the character table of G is very helpful during the investigation of CG (t)-orbits. However, the character table of E7 (2) is unknown at the moment. In fact, since 1987 when Black and Fischer calculated the complex character table of E6 (2), there has been no effort to determine the full character table of the remaining exceptional groups. Consequently, we are unable to compute the permutation rank of G on it is action on X or find the the size of the set XC . Additionally, the GAP code“ClassMultiplicationCoefficient” cannot be utilized. As the smallest matrix representation over GF (2) for G is of dimension 56, the attribute “IsConjugate” does not provide the the desired result. However, the conjugacy classes of both involutions and semisimple elements of G are available. In the next section we will present a statement relating to these classes. Moreover, the dimension of the fix space of elements of even order will be applied to build a set of even order elements with the same dimension of eigenspace. Utilizing the above information we will be able to produce a subset of CG (t)-orbits and check its position in ∆i (t) of C(G, X). Involution of E7 (2) Full details form this section can be found in [7], including a complete list of the conjugacy classes of involutions of E7 (2). The involutions of E7 (2) are written in term of the root system of type E7 . We supply the ordering of the root system of type E7 as follows: 7.4. The Commuting Involution Graphs of E7 (2) 122 Table 7.3: Ordering of the root system of type E7 α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 α14 α15 α16 α17 α18 α19 α20 α21 α22 α23 α24 α25 α26 α27 α28 α29 α30 α31 α32 α33 α34 (1 (0 (0 (0 (0 (0 (0 (1 (0 (0 (0 (0 (0 (1 (0 (0 (0 (0 (0 (1 (1 (0 (0 (0 (0 (1 (1 (0 (0 (0 (0 (1 (1 (1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 0 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 0) 0) 0) 0) 0) 0) 1) 0) 0) 0) 0) 0) 1) 0) 0) 0) 0) 0) 1) 0) 0) 0) 0) 0) 1) 0) 0) 0) 0) 1) 1) 0) 0) 1) Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) α35 α36 α37 α38 α39 α40 α41 α42 α43 α44 α45 α46 α47 α48 α49 α50 α51 α52 α53 α54 α55 α56 α57 α58 α59 α60 α61 α62 α60 = α0 (0 (0 (1 (1 (1 (0 (0 (1 (1 (1 (0 (1 (1 (1 (0 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (1 (2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 1 2 2 2 2 1 1 2 1 1 1 1 2 1 1 1 2 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 3 3 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 3 2 2 3 3 2 3 3 3 3 3 4 4 4 1 1 1 1 1 2 1 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 1 1 0 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 2 2 2 2 2 2 2 123 0) 1) 0) 0) 1) 0) 1) 0) 0) 1) 1) 0) 1) 1) 1) 0) 1) 1) 0) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) Now by identifying xαi with the root subgroup Uαi as defined in [7, 16.1]. Thus we can obtain the structure of the conjugacy classes of involutions in E7 (2) as in the following theorem: Theorem 7.4.1. Let t ∈ E7 (2) be an involution , then t is of the form: i- t1 = xα0 . ii- t2 = xα59 xα58 . iii- t3 = xα53 xα55 xα54 . iv- t4 = xα48 xα47 xα49 . 7.4. The Commuting Involution Graphs of E7 (2) 124 v- t5 = xα53 xα49 xα47 xα48 . Proof. See [7, 16.1] for full details of the proof. The subgroup generated by removing the ith node from the Dynkin diagram is a maximal parabolic subgroups of G denoted by Pi . In the next theorem we give a relationship between the maximal parabolic subgroups of G and the centralizer of these involutions. Theorem 7.4.2. Let t1 , t2 , ...t5 the involutions shown in Theorem 7.3.1. There is a maximal parabolic subgroup Pi of G containing the centralizer of these involutions as described below: i- CG (t1 ) 6 P1 ; ii- CG (t2 ) 6 P6 ; i- CG (t3 ) 6 P3 ; i- CG (t4 ) 6 P7 ; and i- CG (t5 ) 6 P2 , P7 . Proof. See [7, 16.20] for full details of the proof . Let Pi be a maximal parabolic subgroup of G, and denote the maximal normal unipotent subgroup of Pi by Qi and its Levi complement by Li . The following theorem sets up the structure of the centralizer of involution in G. Theorem 7.4.3. The possible structure of CG (t), where t is an involution in G, is as follows: 1. CG (t1 ) = Q1 L1 = P1 . Furthermore, Q1 is an extraspecial 2-group. Thus Q1 /Z(Q1 ) has the structure of an orthogonal space, upon which L1 ∼ = Ω+ 12 (2) acts irreducibly; 2. CG (t2 ) = Q6 L, such that L ∼ = Sp8 (2) × Sym(3); Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 125 3. CG (t3 ) = CQ3 (t3 ) L, such that CQ3 (t3 ) 6 Q3 and L ∼ = Sym(3) × Sp6 (2) 6 L3 ; 4. CG (t4 ) = Q7 L, such that L ∼ = F4 (2) 6 L7 ; 5. CG (t5 ) = Q L, such that Q 6 Q2 Q7 and L ∼ = Sp6 (2) 6 L2 ∩ L7 ∼ = L6 (2). Proof. see [7, 16.20] In conformity with The Atlas giving priority to the larger centralizer, we may name by 2A, 2B, 2C, 2D and 2E the conjugacy classes of the involutions t1 , t2 , t3 , t4 and t5 respectively. It can be shown in [13] that the dimension of the fix space of these involutions is as follows: Table 7.4: Dimension of Fix Space Class 2A 2B 2C 2D 2E dimF ix 44 36 32 28 28 The above result can be used as a tool to distinguish between different conjugacy classes of involutions in G, with exception of the classes 2D and 2E, where they have the same dimension on their fix space. However, we are going to apply the next lemma to differentiate between the classes 2D and 2E. Lemma 7.4.4. Let x and y be involutions of a finite group G . We have that x ∈ y G if the order of x ∗ y is odd. Proof. The basic proof can be seen from [10, 45.1,45.2]. Now let x and y be fixed involutions in the classes 2D and 2E respectively and let z ∈ G be an involution with dimension of fix space equal to 28. In order to test whether z ∈ 2D or z ∈ 2E we check the product x ∗ z g and y ∗ z g such that g runs as a random element of G. Using lemma 7.4.4 we decide in which class z lies by taking the first odd product. 7.4. The Commuting Involution Graphs of E7 (2) 7.4.1 126 Semisimple Classes of E7 (2) Frank Luebeck in [49] showed that the group E7 (2) has 128 conjugacy classes of semisimple elements and his work included the order of the centralizer these elements. Moreover, Peter Rowley and his phd students (J.Ballantyne, C.Bates) in [13] provide the structure of such centralizers. Full details about the semisimple elements of E7 (2) are listed in the following table as shown in [13]. Table 7.5: Semisimple classes of E7 (2) x 3A 3B 3C 3E 5A 7A 7B 7C 9A 9B 9C 9D 9E 9F 11A 13A 15A 15B 15C 15D 15E 15F 15G 17AB 19A 21A 21B 21CD CG (x) 3.2 E6 (2).3 3 × Ω+ 12 (2) 3 × U7 (2) 3. (U3 (2) × U6 (2)).3 5 × Ω− 8 (2) × Sym(3) 7 × L6 (2) 7 × 3 D4 (2) 7 × L3 (2) × L2 (8) 9 × 3 D4 (2) 9 × U5 (2) 9 × U3 (8) 9 × U4 (2) × Sym(3) 9 × L2 (8) × Sym(3) [33 ].2.3.[22 ].[33 ].2 11 × 31+2 : 2.Atl(4) 13 × L2 (8) 15 × Ω− 8 (2) 15 × Alt(8) × Sym(3) 15 × Alt(8) × 3 15 × Alt(5) × (Sym(3))2 5 × 31+2 : 2.Alt(4) × Sym(3) + 15 × Alt(5) × Sym(3) × 3 5 × 31+2 : 2.Alt(4) × 3 + 17 × Alt(5) × Sym(3) 19 × 9 21.L3 (4).3 21 × Alt(8) 21 × L2 (8) Factored Order of CG (x) 236 .311 .52 .72 .11.13.17.19 230 .39 .52 .72 .11.17.31 221 .38 .52 .7.11.17 218 .310 .5.7.11 213 .35 .52 .7.17 215 .34 .5.73 .31 212 .34 .73 .13 26 .33 .73 212 .36 .72 .13 210 .37 .5.11 29 .36 .7.19 27 .37 .5 24 .35 .7 24 .37 23 .34 .11 23 .32 .7.13 212 .35 .52 .7.17 27 .34 .52 .7 26 .34 .52 .7 24 .34 .52 24 .35 .5 23 .34 .52 23 .35 .5 23 .32 .5.17 32 .19 26 .34 .5.72 26 .33 .5.72 23 .33 .72 Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) x 21E 21F 21G 21H 31ABC 33AB 33CDEF GH 33I 35A 39A 43ABC 45AB 51AB 51CD 51EF 57AB 63ABC 63D 63EF G 63HIJ 73ABCD 85ABCD 91ABC 93ABC 99AB 105AB 105CDE 117ABC 127ABCDEF GHI 129ABCDEF 171ABCDEF 217ABCDEF 255ABCD CG (x) 63 × L3 (2) 7 × 31+2 : 2.Alt(4) + 21 × Alt(5) × Sym(3) 7×3×9 31 × L3 (2) 11 × 31+2 : 2.Alt(4) 11 × 31+2 : 2 11 × 31+2 35 × 3 × Sym(3) 117 43 × 3 5 × 9 × Sym(3) 51 × Alt(5) 51 × Sym(3) 51 × 3 19 × 9 63 × L3 (2) 63 × 7 63 × Sym(3) 63 × 3 73 85 × Sym(3) 91 93 99 35 × 3 × Sym(3) 35 × 3 × 3 117 127 129 171 217 255 Factored Order of CG (x) 23 .33 .72 23 .34 .7 23 .33 .5.7 33 .7 23 .3.7.31 23 .34 .11 2.33 .11 33 .11 23 .32 .5.7 32 .13 3.43 2.33 .5 22 .32 .5.17 2.32 .17 32 .17 32 .19 23 .33 .72 32 .72 2.33 .7 33 .7 73 2.3.5.17 91 93 32 .11 2.32 .5.7 32 .5.7 32 .13 127 3.43 32 .19 7.31 3.5.17 127 7.4. The Commuting Involution Graphs of E7 (2) 128 As we see, the exceptional group of Lie-type E7 (2) has 5 conjugacy classes of involution. Our aim is to study the commuting involution graphs for the classes 2A, 2B and 2C by finding a subset of XC with size equal or nearly equal to the size of XC . To deal with the difficulties that arise during our search for CG (t)-orbits where t is an involution in one of our aforementioned classes, we produce the algorithms given in the next subsection. 7.4.2 Algorithms Let z ∈ G of order m and C = z G . The purpose of the first algorithm is to find a subset of elements from the nonempty set XC . This algorithm is based on the results of Lemma 7.2.2. We shall also note that the Magma code “UnipotentStabilizer” and “Eigenspace” are involved. The algorithm is as follows: Algorithm 5 . Input: let t be an involution in E7 (2), and z ∈ G of order m and C = z G ; i: d −→ Dimension(Eigenspace(z, 1)),O −→ M ultiples(m)∩ Divisors of |G|. ii: repeat r −→ Random(tG ),o −→ order(t ∗ r). iii: if o ∈ O, then s −→ IntegerRing()!(o/m) . vii: stop when Dimension(Eigenspace((t ∗ r)s , 1)) equal d. iv: ORB −→ {t ∗ (t ∗ r)s } v: In order to get new orbits repeat the steps from ii to iv and let a2 be the involution obtained in step iv and a1 ∈ ORB. They will be different if one of the following holds vi: #U nipotentStabilizer(O2 , Eigenspace(a1 , 1))) 6= #U nipotentStabilizer(O2 , Eigenspace(a2 , 1))) Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 129 vii: Dimension(Eigenspace(t, 1) ∩ Eigenspace(a1 , 1) 6= Dimension(Eigenspace(t, 1) ∩ Eigenspace(a2 , 1) viii: ORB −→ ORB ∪ {a2 } Output: the set of suborbit representatives with same order and same dimension of fix space. As a matter of fact, for g ∈ G, the stabilizer of F ixg in G contains the centralizer in G of g. Therefore, we may apply this fact to check whether the elements in CG(t) -orbits are in ∆2 (t). The Magma codes “Eigenspace”, “LMGSylow” and “UnipotentStabilizer” are used in this work. The following algorithm serves this purpose: .Algorithm 6 Input: let t be an involution in E7 (2) and let b ∈ CG (t)-orbits, with order(t∗b) > 2; i: E −→ (Eigenspace(b, 1)). ii: Sy −→ LM GSylow(CG (t), 2), U N −→ U nipotentStabilizer(Sy, E). iii: if there is y ∈ U N such that (b, y) = (y, b) = Id(G) then b ∈ ∆2 (t). iv: if not take Sy −→ Sy Random(CG (t)) and repeat the steps i and ii and check if step iii holds. Output: the set of suborbit representatives belonging to ∆2 (t). The purpose of this algorithm is to locate a sufficient number of subgroups of the centralizer in G of a semisimple element x ∈ G. The centralizer will then be the subgroup generated by all such subgroups. The code “RandomElementOfOrder”, 7.4. The Commuting Involution Graphs of E7 (2) 130 “CompositionFactors” and “LMGcentralizer” contributed to the process of building this algorithm. We can also find the size of the CG (x) from Table 7.5. The following algorithm is as follows: .Algorithm 7 Input: let x be a semisimple element of G; i: CG (x) = {Id(G)}, repeat z −→ RandomElementOf Order(G, 2). ii: U −→ sub < GL(56, 2)|z, x >. iii: M F −→ CopositionF actors(GM odule(U )). iv: if size of M F greater than or equal to 20, then S −→ LM Gcentralizer(U, x). iv: CG (x) −→ sub < GL(56, 2)|CG (x), S >. vi: repeat the above steps until we have sufficient numbers of subgroups of CG (x) to generate the centralizer of x in G. Output: CG (x) Definition 7.4.5. [32] Let G be a finite soluble group. A presentation for G of the k form < s1 , s2 , . . . , sn |sj j = hjj , 1 6 j 6 n, ssj i = hij , 1 6 i < j 6 n >, where k k 1- kj is the least prime such that sj j ∈< sj+1 , sj+2 , . . . , sn > for j < n, and sj j = IdG if j = n, and 2- hij is a word in the generators for si+1 , si+2 , . . . , sn , will be called a power-conjugate presentation (pc-presentation) for G. Now, a Sylow 2-subgroup of a finite group G contains representatives of all the conjugacy classes of involutions of G. Thus we create the next algorithm to compute a subset of CG (t)-orbits belonging to ∆1 (t). To do this we firstly need to calculate a Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 131 Sylow 2-subgroup of CG (t) by using the Magma code “LMGSylow(CG (t),2)”. We then turn it into a pc-group using the code “LMGSolubleRadical”. This makes it possible to obtain a set of representatives of a given class of involutions belonging to the Sylow 2-subgroup of CG (t). By checking the action of CG (t) on this set by conjugation we get the ∆1 (t). The Magma code “ConjugacyClasses”, “LMGIsIn” and “LMGIsConjugate” are applied in this work. The algorithm is as follows: Algorithm 8 . Input: let t be an involution in E7 (2), C = CE7 (2) (t); i: S −→ LM GSylow(C, 2). ii: N1 , N2 , N3 −→ LM GSolubleRadical(S), N1 the soluble radical of S, N1 ∼ = N2 ∼ = S, N2 is pc-group and N3 isomorphism from S to N2 . iii: D −→ ConjugacyClasses(N 2). iv: let K be a set of representative of conjugacy classes of involution in S. We move them into representatives of conjugacy classes of involution in C. v: a −→ Random(K). vi: repeat c −→ Random(C). vii: if LM GIsIn(S c , a) then T1 , T2 , T3 −→ LM GSolubleRadical(S c ). viii: for x in K check if LM GIsConjugate(T2 , T3 (xc ), T3 (a)) then K −→ K \ {x} Output: small set of representative of conjugacy classes of involution in ∆1 (t). We are going to utilise all of the above algorithms, in addition to the ones in previous chapters to investigate the commuting involution graphs. 7.4. The Commuting Involution Graphs of E7 (2) 7.4.3 132 Disks Structure and Orbit Size For t be an involution in classes 2A, 2B or 2C in E7 (2). Let X = tE7 (2) , so that for x ∈ X and m equal to order(t ∗ x), then by Theorem 4.2.2 part 2 if m = 2, then x ∈ ∆1 (t). Moreover, if m is even, m > 4 and (t ∗ x)m/2 ∈ X, then x ∈ ∆2 (t). Otherwise we may employ Algorithm 6 to show that x ∈ ∆2 (t) or Algorithm 4 to prove that x ∈ ∆3 (t). To calculate ∆1 (t) we first apply Algorithm 8 to get a small set of representatives of involutions in CG (t), then we used Lemma 7.2.2 and Lemma 7.2.3 to obtain a smaller set which allows us to break it into CG (t)-orbits by the Magma code “Isconjugate”. Therefore we find the complete orbits of ∆1 (t). On the other hand, to calculate the size of the orbit containing x it is enough to find the CCG (t) (x). This is because the size of the CG (t)-orbit with representative x is equal |CG (t)| . However, to compute CCG (t) (x) we consider the following subcases : to |CCG (t) (x)| 1. If m = 2k and k is even, then we apply the following procedures to calculate CCG (t) (x): i- Compute the centralizer of the involution w1 = (t ∗ x)k inside CG (t) by using “LMGCentralizer” Magma code. call this centralizer W1 . ii- Calculate CW1 (w2 ),such the w2 = t ∗ x of order 2k. This can be done by using the code “ApproximateStabilizer” to find the stabilizer in W1 for the fix space of w2 namely StW1 (w2 ). We then apply the “OrbitBounded” code to confirm the size of the stabilizer. Moreover, it is a straight forward check to see that the generator set of StW1 (w2 ) commutes with w2 this leads to CW1 (w2 ) = StW1 (w2 ). Set W3 = CW1 (w2 ). iii- We find CW3 (x) by employing the codes “ApproximateStabilizer” and “OrbitBounded” respectively. By Theorem 4.2.2 part 1 we have CCG (t) (x) = CW3 (x). 2. If m = 2k and k is odd, then to find CCG (t) (x) we use the following steps: Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 133 i- Use Algorithm 7 to calculate the centralizer of the semisimple element w1 = (t ∗ x)2 in G. Let W1 = CG (w1 ). ii- Compute CW1 (w2 ) where w2 = t ∗ x using the “LMGRadicalQuotient” code. We may let W3 = CW1 (w2 ). iii- Again we employ the code “ApproximateStabilizer” to calculate CCG (t) (x) = CW3 (x). 3. If m is odd, then in a similar way as case 2 we can calculate CCG (t) (x), but with the advantage that we need only apply the parts ii and iii. 7.4.4 Connectivity According to the online Atlas we may generate G = E7 (2) by two element x, y such that x has order 5 and y has order 6. By Theorem 7.4.1 we know that G has five G-conjugacy classes of involution. For t ∈ G an involution, to show the connectivity of the graph we only need to prove that tx , ty ∈ CC . The reason for this is that for any w ∈ CC there is path between t and w, so conjugating this path by x, y leads to two new paths from tx to wx and from ty to wy . Here CC = StG (CC ) and the connectivity of the graph follows by Lemma 3.5.1. The connectivity of C(E7 (2), 2A) follows by using a random search to detect an involution t ∈ 2A such that tx , ty ∈ ∆1 (t). On the other hand, one can find t ∈ 2B with properties tx ∈ ∆1 (t) and using Theorem 4.2.2 part 3 we can confirm that ty ∈ ∆2 (t). Hence C(E7 (2), 2B) is connected. Moreover, choosing a random t ∈ 2C and applying Theorem 4.2.2 part 3 we get tx ∈ ∆2 (t) and Algorithm 4 shows that ty ∈ ∆3 (t). Therefore, the graph C(E7 (2), 2C) is connected. Finally, seeking an involution t ∈ 2D or 2E and applying Algorithm 6 we get tx ∈ ∆2 (t). On the other hand, there are x1 , x2 ∈ CG (ty ) conjugate to t satisfies (t, x1 ) = (x1 , x2 ) = (x2 , ty ) = 1. Consequently, tx , ty ∈ CC and the connectivity of the graph is proved. 7.5. Commuting Involution Graph of 2A 7.5 134 Commuting Involution Graph of 2A Let t be an involution in class 2A. We have the following: 1- t = xα0 . 2- The dimension of the fix space of t is 44. 3- The size of X = tG is equal to 18610317999. 4- CG (t) = P1 . 5- The 2-core of CG (t), namely O2 , has size 233 . Our strategy is to compute the CG (t)-orbit by computing the set XC . This is because a non-empty set XC breaks down into CG (t)-orbits. Therefore, we are looking for the CG (t)-orbit inside the non-empty set XC . Moreover, for x is a representative of CG (t)-orbit, as O2 is a unipotent subgroup then the Magma code “UnipotentStabilizer”, is applicable to calculate the stabilizer of the fix space of x inside the 2-group O2 , we name that by UO2 (x) and we used it as a CG (t)-orbit invariant (see Lemma 7.2.2 part ii). However, the code “ApproximateStabilizer” will be utilized to compute the size of the orbits. We will now apply the above ideas as follows: X2A By using Algorithm 5 there is an involution x ∈ X with t ∗ x ∈ 2A. We use the code “ApproximateStabilizer” to find the Stabilizer in CG (t) for the fix space of x, namely StCG (t) (x), and the “OrbitBounded” code to confirm the size of the Stabilizer. By a straight forward check one can see that the generator set of StCG (t) (x) commutes with x, and here CCG (t) (x) ∼ = StCG (t) (x). The size of this orbit is equal to 151470 and |UO2 (x)| = 4294967296. Theorem 4.2.2 part 2 then shows that x ∈ ∆1 (t). X2B Algorithm 5 shows that there is x ∈ X such that t ∗ x ∈ 2B. In order to find CCG (t) (x) we apply the “LMGRadicalQuotient” code on CG (t). This will give an epimorphism from CG (t) to a permutation group isomorphic to the quotient Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 135 group CG (t)/O2 . Let W1 be the centralizer of the image of x and W2 to be the inverse image of W1 inside CG (t) so that CCG (t) (x) 6 W2 . The code “ApproximateStabilizer” provides us with StW2 (x) and by checking the generators of SW2 (x) we obtain CCG (t) (x) ∼ = StW2 (x) and |StW2 (x)| = 4596373779694328217600. Additionally, using the above technique we get |UO2 (x)| = 33554432 and x ∈ ∆1 (t). X3B By applying Algorithm 5 we obtain x ∈ X such that t ∗ x ∈ 3B. From table 7.5 and Algorithm 7 we calculate CG (t ∗ x) and by Theorem 4.2.2 part 1 we obtain CCG (t) (x). We find that the size of this orbit is 8589934592 and |UO2 (x)| = 1. Moreover, Algorithm 7 shows that x ∈ ∆2 (t). X4 In this case Algorithm 5 gives an element x ∈ X with order(t ∗ x) = 4 and the dimension of the fix space of t ∗ x is 32. To compute CCG (t) (x), let z = (t ∗ x)2 . By checking the dimension of the fix space of z we obtain z ∈ 2A, and we find CCG (t) (z) as above. Inside this group we apply the code “ApproximateStabilizer” to calculate CCG (t) (x). Therefore, we have the size of this orbit is 9926737920 and |UO2 (x)| = 65536, and x ∈ ∆2 (t) by Theorem 4.2.2 part 3. From the above cases we see that a complete list of CG (t)-orbits are found, so that we can establish the following theorem: Theorem 7.5.1. Let G be the exceptional group E7 (2). Suppose that t ∈ 2A. We have DiamC(E7 (2), 2A) = 2 with |∆1 | = 93645486 and |∆2 | = 18516672512. Proof. From the above we can pin down the following table Class Or Order |F ix(t∗x) | |UO2 (x)| ∆1 (t) ∆2 (t) 2A 44 4294967296 151470 − 2B 36 33554432 93494016 − 3B 32 1 − 8589934592 4 32 65536 − 9926737920 7.6. Commuting Involution Graph of 2B 7.6 136 Commuting Involution Graph of 2B Let t be an involution in class 2B. We have the following: 1- t = xα59 xα58 . 2- The dimension of the fix space of t is 36. 3- The size of X = tG is equal to 6396887385160272. 4- CG (t) = Q6 L, where L = Sp8 (2) × Sym(3). 5- The 2-core of CG (t), namely O2 , has size 242 . Computing the non-empty XC for the class 2B requiring a lot more time and effort than we had in case the class 2A, besides we can not cover the complete list of CG (t)orbits. Thus we attempt as much as we can to collect the largest possible number of CG (t)-orbits. Since no enough information about the conjugacy classes for the elements of even order greater than 2, we let any two element of even order greater than 2 be conjugate in G if they have the same order and the same dimension of fix space. For x a random element in the set XC we calculate the size of the orbit represented by x and determine which ∆i (t) it is located in. C Is Involution Class We will start first with the sets XC such that C is a conjugacy class of involution in G where XC lies in ∆1 (t). For x an involution in such sets the code “LMGcentralizer” will be employed to calculate its centralizer inside CG (t). This is possible because x ∈ CG (t) and |CG (t)| is small enough to allow as to compute with it. Furthermore, The method we employ to determine ∆1 (t) is established in section 7.4.3. This method leads to the size of ∆1 (t), which is equal to 19113646671, and with 17 suborbits of CG (t) . The next table explains the above statements: Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 137 Class Dim(F ix(t∗x) ) Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆1 (t) 2A 44 34 4398046511104 255 2A 44 30 137438953472 220320 2B 36 34 4398046511104 240 2B 36 29 137438953472 6609600 2B 36 26 4294967296 109670400 2B 36 27 8589934592 47001600 2B 36 26 67108864 53477376 2B 36 26 16777216 66846720 2B 36 26 3435973868 8812800 2C 32 26 4294967296 82252800 2C 32 26 3435973868 82252800 2C 32 24 16777216 4211343360 2C 32 24 536870912 5922201600 2D 28 26 67108864 50135040 2D 28 26 8589934592 50135040 2E 28 22 67108864 4211343360 2E 28 22 16777217 4211343360 7.6. Commuting Involution Graph of 2B 138 C Is Class of Order 3 Let x be an involution in 2B such that order(t ∗ x) = 3. There are just two such orbits and they are in ∆2 (t) by Algorithm 6. The are as follows: Class Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆2 (t) 3B 24 131072 9126805504 3D 18 1 4398046511104 C Is Class of Order 4 We investigate the sets XC such that C is a class of elements of order 4. Now for x ∈ X such that t ∗ x ∈ C Algorithm 6 shows that x ∈ ∆2 (t). By considering the dimension of the fixed space for (t ∗ x), the CG (t)-orbits break up as follows: (t ∗ x)2 − class 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2B 2B 2B 2B 2B 2C 2B 2B Dim(F ix(t∗x) ) 32 32 32 32 26 26 26 26 26 26 26 26 24 24 24 24 24 20 20 18 18 Dim(F ixt ∩ F ixx ) 24 24 24 26 21 21 21 21 22 22 24 24 20 24 20 20 24 18 18 17 17 |UO2 (x)| 16777216 33554432 536870912 4294967296 16777216 2097152 8388608 1048576 8388608 67108864 536870912 33554432 1048576 131072 65536 1048576 16777216 65536 8192 65536 512 ∆2 (t) 3743416320 14438891520 6768230400 658022400 14438891520 433166745600 433166745600 433166745600 126340300800 126340300800 13536460800 13536460800 6064334438400 8556380160 6930667929600 6930667929600 8556380160 43124156006400 172496624025600 6571299962880 6571299962880 Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 139 C Is Class of Order 5 Suppose that x ∈ CG (t)-orbits such that order(t ∗ x) = 5. There is only one such orbits and it is in ∆2 (t) by Algorithm 6, and is described as below: Class Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆2 (t) 5A 16 1 1055531162664960 C Is Class of Order 6 We now look at the sets XC such that C is a class of elements of order 6. Now for x ∈ X such that t ∗ x ∈ C Algorithm 6 shows that x ∈ ∆2 (t). The set XC breaks up into CG (t)-orbits according to the dimension of the fixed space as follows: (t ∗ x)3 class 2A 2B 2B 2B (t ∗ x)2 class 3B 3B 3B 3D Dim (F ix(t∗x) ) 24 20 16 16 Dim(F ixt ∩F ixx ) 20 18 16 16 |UO2 (x)| ∆2 (t) 65536 8192 512 1 7392712458240 229995498700800 7009386627072 1121501860331520 C Is Class of Order 8 Let x be in the set XC such that C is a class of elements of order 8 and dimension of fix space equal 16. Then we have (t ∗ x)4 is conjugate to 2A , and Dim(F ix(t∗x)2 ) = 26. In this case the CG (t)-orbits are described as follows: Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆2 (t) 16 256 1774250989977600 C Is Class of Order 12 In this case we are looking for the set XC such that C are a class of elements of order 12 and the dimension of the fix space is 16. For x ∈ X such that order(t ∗ x) = 12 7.7. Commuting Involution Graph of 2C 140 we have (t ∗ x)6 is conjugate to 2A , (t ∗ x)4 is conjugate to 3B, Dim(F ix(t∗x)3 ) = 32, Dim(F ix(t∗x)2 ) = 24 and x ∈ ∆2 (t) by Algorithm 6. To calculate CG (t) we use Algorithm 7 to compute CCG (t) ((t ∗ x)6 ) and inside this group we find CG (t) by using “ApproximateStabilizer” code. The CG (t)-orbits are as follows: 7.7 Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆2 (t) 16 256 1892534389309440 Commuting Involution Graph of 2C Let t be an involution in class 2C. We have the following: 1- t = xα53 xα55 xα54 . 2- The dimension of the fix space of t is 32. 3- The size of X = tG is equal to 26099300531453909760. 4- CG (t) = CQ3 (t3 ) L, such that CQ3 (t3 ) 6 Q3 and L = Sym(3) × Sp6 (2) 6 L3 . 5- The 2-core of CG (t), namely O2 , has size 245 . Our target is to collect the largest subset of CG (t)-orbits for the C(G, 2C). Moreover, the research involves determining the size of the CG (t)-orbits and the place of the orbits in ∆i (t). Moreover, we note that Algorithms 3, 4, 5, 6, 7 are involved in this calculation . We should note that the deal with sets XC such C is class of elements of even order greater than 2 is the same as the case the class 2B. The set XC breaks up between the CG (t)-orbits as described below: C Is Involution Class Let C be a conjugacy class of involution in G. For x in XC such set order(t ∗ x) = 2, we have x ∈ ∆1 (t). Moreover, the procedure give in in Section 7.4.3 will cover all CG (t)orbits when C is an involution class. As a result of this we can figure out ∆1 (t). This Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 141 technique produces the size of ∆1 (t) which is equal to 49116616055 with 74 suborbits of CG (t). Full information about these orbits is included in the following table: Class Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆1 (t) 2A 26 549755813888 120960 2A 26 2147483648 98304 2A 32 35184372088832 3 2A 28 35184372088832 1260 2B 26 35184372088832 15120 2B 26 35184372088832 16128 2B 26 549755813888 120960 2B 23 549755813888 8709120 2B 22 34359738368 30965760 2B 22 34359738368 23224320 2B 22 2147483648 33030144 2B 22 134217728 82575360 2B 22 8589934592 20643840 2B 22 2147483648 30965760 2B 23 68719476736 11612160 2B 24 68719476736 1032192 2B 24 549755813888 724760 7.7. Commuting Involution Graph of 2C 142 Class Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆1 (t) 2C 20 17179869184 278691840 2C 20 2147483648 283115520 2C 20 2147483648 283115520 2C 20 2147483648 283115520 2C 20 2147483648 283115520 2C 20 2147483648 396361728 2C 20 2147483648 396361728 2C 20 2147483648 396361728 2C 20 2147483648 396361728 2C 20 67108864 2972712960 2C 20 67108864 2972712960 2C 20 67108864 990904320 2C 20 16777216 3963617280 2C 26 35184372088832 11520 2C 26 35184372088832 11520 2C 22 8589934592 92897280 2C 22 8589934592 247726080 2C 22 536870912 247726080 2C 20 536870912 990904320 2C 22 68719476736 69672960 Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) Class Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆1 (t) 2C 20 2147483648 1114767360 2C 22 134217728 247726080 2C 20 134217728 990904320 2C 20 2147483648 495452160 2C 20 17179869184 371589120 2C 22 549755813888 17418240 2C 22 68719476736 23224320 2C 22 549755813888 23224320 2C 20 4294967296 495452160 2C 20 4294967296 371589120 2C 20 17179869184 371589120 2C 20 17179869184 495452160 2C 24 549755813888 483840 2C 22 34359738368 23224320 2C 20 2147483648 371589120 2C 22 68719476736 30965760 2C 22 34359738368 30965760 2C 22 34359738368 30965760 2C 22 34359738368 30965760 2C 24 68719476736 1032192 2D 20 68719476736 15482880 2D 20 8589934592 15482880 2E 18 67108864 1981808640 143 7.7. Commuting Involution Graph of 2C 144 Class Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆1 (t) 2E 18 536870912 1486356480 2E 18 16777216 11890851840 2E 18 268435456 1981808640 2E 18 268435456 1486356480 2E 20 4294967296 92897280 2E 20 68719476736 92897280 2E 20 67108864 2972712960 2E 20 67108864 99094320 2E 20 67108864 2972712960 2E 20 536870912 99094320 2E 20 536870912 743178240 2E 20 8589934592 743178240 2E 20 8589934592 557383680 2E 20 2147483648 557383680 C Is Class of Order 3 We consider the sets XC when C is a conjugacy class of elements of order 3. The distribution of this set is as follows: Class Dim(F ixt ∩ F ixx ) |UO2 (x)| ∆2 (t) 3B 20 32768 6442450944 3B 20 16777216 2818572288 3E 14 1 35184372088832 3D 14 2048 5772436045824 Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 145 C Is Class of Order 4 We now look at the sets XC where C is a conjugacy class of elements of order 4 which have the same dimension of fix space. Full information about the size of these orbits and their position in the ∆i (t) is drawn up in the following table: (t ∗ x)2 − class 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2B 2B 2B 2A 2B 2B 2A 2A 2A 2A Dim(F ix(t∗x) ) 32 32 32 32 32 26 26 26 26 26 26 26 26 26 26 26 26 26 24 24 24 24 24 24 24 24 24 22 Dim(F ixt ∩ F ixx ) 20 20 20 20 22 17 17 17 17 17 17 18 18 18 18 19 19 20 16 16 16 16 16 16 16 17 18 16 |UO2 (x)| 16777216 536870912 33554432 32768 549755813888 8388608 4194304 33554432 67108864 16777216 134217728 33554432 536870912 2147483648 134217728 2147483648 134217728 4294967296 524288 524288 1948576 32768 8388608 2097152 4194304 16777216 134217728 4194304 ∆2 (t) 3963617280 3963617280 3963617280 3221225472 3221225472 285380444160 285380444160 142690222080 142690222080 142690222080 23781703680 23781703680 17836277760 17836277760 23781703680 4459069440 11890851840 743178240 2029372047360 2029372047360 2029372047360 2029372047360 2283043553280 2283043553280 1141521776640 285380444160 23781703680 1141521776640 7.7. Commuting Involution Graph of 2C (t ∗ x)2 − class 2B 2B 2B 2B 2B 2C 2C 2B 2B 2C 2B 2B 2B 2C 2B 2B 2B 2B 2B 2C 2C 2C 2C 2C 2C 2C Dim(F ix(t∗x) ) 20 20 20 20 20 20 20 20 20 20 20 20 20 20 18 18 18 18 18 16 16 16 16 16 16 16 146 Dim(F ixt ∩ F ixx ) 14 14 14 14 14 14 14 14 14 15 16 16 18 18 13 13 13 13 14 12 12 12 12 12 13 14 |UO2 (x)| 16384 32768 32768 524288 1048576 512 65536 2097152 131072 524288 8388608 131072 134217728 67108864 32768 65536 2097152 262144 4194304 16384 8192 4096 4096 512 65536 524288 ∆2 (t) 24352464568320 24352464568320 16234976378880 16234976378880 6088116142080 64939905515520 32469952757760 1522029035520 18264348426240 18264348426240 1141521776640 1014686023680 31708938240 42278584320 48704929136640 48704929136640 2029372047360 2029372047360 1522029035520 97409858273280 97409858273280 292229574819840 292229574819840 389639433093120 73057393704960 24352464568320 C Is Class of Order 5 We investigate the set XC such that C is a conjugacy class of elements of order 5. We only found two such orbits and details about them are listed in the following table: Class 5A 5A Dim(F ixt ∩ F ixx ) 12 12 |UO2 (x)| 64 2048 ∆2 (t) 1108307720798208 277076930199552 Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 147 C Is Class of Order 6 We examine the sets XC with C a conjugacy class of elements of order 6, with the property that they have the same dimension of the fix space. For an orbit representative x in such sets we will determine the size of the orbit of x and it is place within the ∆i (t). Details are given in the table below: (t ∗ x)3 class 2C 2C 2C 2C 2C 2A 2A 2B 2B 2C 2C 2A 2B 2B 2B 2B 2B 2B 2A 2A 2A 2A (t ∗ x)2 class 3E 3D 3D 3D 3D 3D 3D 3D 3D 3B 3B 3E 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B Dim (F ix(t∗x) ) 12 12 12 12 12 16 16 16 16 16 16 20 20 20 20 20 20 20 24 24 24 24 Dim (F ixt ∩ F ixx ) 10 10 10 10 12 12 12 12 14 12 12 14 14 14 14 14 14 15 16 16 16 18 |UO2 (x)| ∆2 (t) 1 64 128 512 1024 1024 64 1204 2048 512 16384 1 512 32768 65536 16384 524288 2097152 2097152 65536 32768 16777216 33249231623946240 16624615811973120 12468461858979840 12468461858979840 1039038488248320 519519244124160 554153860399104 1039038488248320 17317308137472 389639433093120 129879811031040 105553116266496 129879811031040 97409858273280 43293270343680 43293270343680 24352464568320 18264348426240 3044058071040 2164663517184 2029372047360 42278584320 C Is Class of Order 7 We analyze the set XC such that C is a conjugacy class of elements of order 7. For x in such a set the information on the orbits is included in the following table: Class 7A 7C Dim(F ixt ∩ F ixx ) 14 8 |UO2 (x)| 1 1 ∆3 (t) 211106232532992 10133991615836160 7.7. Commuting Involution Graph of 2C 148 C Is Class of Order 8 Here we check the sets XC with C a conjugacy class of elements of order 8 having the same dimension of fix space. For an element x in the set, we will determine the size of the orbit of x and the position of x in the ∆i (t), as shown in the following table: (t ∗ x)4 class 2C 2C 2C 2C 2B 2B 2B 2B 2B 2B 2B 2C 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A Dim Dim Dim (F ix(t∗x) ) (F ix(t∗x)2 ) (F ixt ∩ F ixx ) 8 16 8 8 16 8 10 20 9 10 20 9 12 20 10 12 20 10 12 20 10 12 20 10 12 20 10 12 20 10 12 20 12 12 20 10 14 26 11 14 26 11 14 26 11 14 24 11 16 26 12 16 26 12 16 26 12 16 26 12 16 26 12 16 26 12 20 32 14 20 32 14 20 32 14 20 32 14 |UO2 (x)| ∆2 (t) 16 32 256 32 32 512 64 1024 128 256 8192 2048 512 2048 8192 4096 32768 16384 4096 512 8192 131072 1024 512 8192 65536 14962542307758080 4987384735919360 4987384735919360 4987384735919360 12468461858979840 12468461858979840 12468461858979840 12468461858979840 9351346394234880 9351346394234880 779278866186240 18702692788469760 4675673197117440 2337836598558720 2337836598558720 779278866186240 58445914963960 58445914963960 58445914963960 259759622062080 389639433093120 146114787409920 129879811031040 129879811031040 129879811031040 32469952757760 Chapter 7. Commuting Involution Graph of 2 E6 (2) and E7 (2) 149 C Is Class of Order 9 In this case we study the set XC such that C is a conjugacy class of elements of order 9. For x a representative on such an orbit important details are listed in the following table: (t ∗ x)-class 9D 9E (t ∗ x)3 −class 3E 3E Dim(F ixt ∩ F ixx ) 10 8 |UO2 (x)| 1 1 ∆3 (t) 709316941310853312 303992974847508480 C Is Class of Order 10 We regard the sets XC with C a conjugacy class of elements of order 10 with equal dimension of fix space. For an orbit representative x in these sets we calculate the size of the orbit of x and the location of x in the ∆i (t), as shown in the table below: (t ∗ x)5 -class 2C 2C 2A 2B 2B 2B Dim(F ix(t∗x) 8 8 12 12 12 12 Dim(F ixt ∩ F ixx ) 8 8 10 10 10 10 |UO2 (x)| 16 32 512 32 64 128 ∆2 (t) 199495389743677440 66498463247892480 8312307905986560 66498463247892480 13299692649578496 16624615811973120 C Is Class of Order 12 We discuss the sets XC with C a conjugacy class of elements of order 12 with equal dimension of fix space. For an element x in such sets we calculate the size of the orbit of x and the location of x in the ∆i (t). This is summarized in table below: 7.7. Commuting Involution Graph of 2C (t ∗ x)6 class (t ∗ x)4 class 2B 2B 2C 2B 2B 2A 2A 2B 2B 2B 2B 2B 2B 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A 3E 3D 3D 3B 3B 3D 3E 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3B 3E 150 Dim Dim Dim Dim (F ix(t∗x) ) (F ix(t∗x)2 ) (F ix(t∗x)3 ) (F ixt ∩ F ixx ) 8 12 20 8 8 16 20 8 8 12 20 8 10 20 18 9 10 20 18 9 12 16 32 10 12 20 24 10 12 20 20 10 12 20 20 10 12 20 20 10 12 20 24 10 12 20 20 10 12 20 24 10 14 24 26 11 14 24 26 11 14 24 26 11 14 24 26 11 14 24 26 11 14 24 26 12 14 24 26 12 14 24 26 12 16 24 32 12 16 24 32 12 16 24 32 12 16 24 32 12 16 24 32 12 20 20 32 14 |UO2 (x)| ∆2 (t) 1 32 16 32 256 64 1 512 32 256 256 128 1024 512 2048 4096 8192 1024 16384 131072 32768 32768 4096 131072 16384 512 1 398990779487354880 66498463247892480 398990779487354880 4987384735919360 16624615811973120 22166154415964160 33249231623946240 12468461858979840 24936923717959680 37405385576939520 18702692788469760 18702692788469760 12468461858979840 4675673197117440 4675673197117440 4675673197117440 4675673197117440 4675673197117440 779278866186240 58445914963960 58445914963960 779278866186240 779278866186240 2922295748119840 86586540687360 259759622062080 105553116266496 C Is Class of Order 13 We contemplate the set XC such that C is a conjugacy class of elements of order 13. Information about the CG (t)-orbit of x can be seen in the table below: Class 13A Dim(F ixt ∩ F ixx ) 8 |UO2 (x)| 1 ∆3 (t) 607985949695016960 Commuting Involution Graph of 2 E6 (2) and E7 (2) Chapter 7. 151 C Is Class of Order 14 We consider the sets XC with C a conjugacy class of elements of order 14 and having the same dimension of fix space. For an orbit representative x in such sets we compute the size of the orbit of x and in which ∆i (t) the element x sitting. The following table describes our findings: (t ∗ x)7 class (t ∗ x)2 class Dim (F ix(t∗x) ) 2A 2B 7C 7A 8 12 Dim (F ixt ∩ F ixx ) 8 10 |UO2 (x)| ∆3 (t) 1 1 303992974847508480 66498463247892480 C Is Class of Order 15 We search for the sets XC such that C is a conjugacy class of elements of order 15. For x in such sets, we provide essential details on the orbits in the following table: (t ∗ x)-class 15A 15B (t ∗ x)5 -class 3B 3D Dim(F ixt ∩ F ixx ) 8 10 |UO2 (x)| 1 1 ∆3 (t) 425590164786511872 70931694131085312 C Is Class of Order 16 We looking for the sets XC with C a conjugacy class of elements of order 16 with equal dimension of fix space. For an element x in such sets we determine the size of the orbit of x and its position in the ∆i (t), as described in the following table : (t ∗ x)8 class 2B 2B 2A Dim Dim Dim Dim (F ix(t∗x) ) (F ix(t∗x)2 ) (F ix(t∗x)4 F ixt ∩ F ixx ) 8 12 20 8 8 12 20 8 8 14 20 8 |UO2 (x)| |∆3 (t)| 4 8 16 398990779487354880 398990779487354880 299243084615516160 7.7. Commuting Involution Graph of 2C 152 C Is Class of Order 17 We investigate the sets XC such that C is a conjugacy class of elements of order 17. For x in such sets, details about the orbits are listed in the following table: (t ∗ x)-class 17A Dim(F ixt ∩ F ixx ) 8 |UO2 (x)| 1 ∆3 (t) 851180329573023744 C Is Class of Order 18 We explore the sets XC with C a conjugacy class of elements of order 18 having the same dimension of fix space. For an orbit representative x in such sets we calculate the size of the orbit of x and determine in which ∆i (t) it is sitting, as shown in the table below: (t ∗ x)9 class (t ∗ x)6 class (t ∗ x)2 class Dim F ix(t∗x)3 2A 3E 9E 20 Dim F ixt ∩ F ixx ) 8 |UO2 (x)| ∆3 (t) 1 303992974847508480 C Is Class of Order 20 We now look at the sets XC with C a conjugacy class of elements of order 20 with equal dimension of fix space. For an element x in such sets we compute the size of the orbit of x and identify the ∆i (t) containing this orbit. Our findings are summarized in the table below: (t∗x)10 class Dim F ix(t∗x) Dim F ix(t∗x)2 Dim F ix(t∗x)5 2A 2B 2B 8 8 8 12 12 12 32 24 24 Dim F ixt ∩ F ixx )| 8 8 8 |UO2 (x)| ∆2 (t) 16 8 4 132996926495784960 53198770583139840 53198770583139840 Commuting Involution Graph of 2 E6 (2) and E7 (2) Chapter 7. 153 C Is Class of Order 21 In this case we investigate the set XC such that C is a conjugacy class of elements of order 21. For x in such orbits, noteworthy details are recorded in the following table: (t ∗ x) -class (t ∗ x)3 class (t ∗ x)7 class 21G 21D 7A 7C 3D 3B Dim (F ixt ∩ F ixx ) 8 8 |UO2 (x)| ∆3 (t) 1 1 851180329573023744 607985949695016960 C Is Class of Order 24 we examine the sets XC with C a conjugacy class of elements of order 24 with dimension of fix space equal 8. We notice that for an orbit representative x in such sets the dimension(F ixt ∩ F ixx ) = 8. The following table describes information about this type of set: (t ∗ x)8 -class 3B 3D 3B (t∗x)12 -class 2A 2A 2B Dim F ix(t∗x)6 26 32 20 Dim F ix(t∗x)4 24 16 20 Dim F ix(t∗x)3 16 20 12 Dim F ix(t∗x)2 14 12 12 |UO2 (x)| ∆i 16 18 4 Orbit Size ∆2 (t) 299243084615516160 ∆2 (t) 53198770583139840 ∆3 (t) 398990779487354880 C Is Class of Order 28 We look at the sets XC with C a conjugacy class of elements of order 28 with dimension of fix space equal 8. For an element x in such sets the dimension(F ixt ∩ F ixx ) = 8. In the following table we include details on such orbits : (t∗x)14 class 2A 2B (t ∗ x)4 class 7C 7A Dim(F ix(t∗x)7 ) Dim(F ix(t∗x)2 ) |UO2 (x)| ∆3 (t) 32 20 8 12 1 1 303992974847508480 797981558974709760 7.7. Commuting Involution Graph of 2C 154 C Is Class of Order 30 We consider the sets XC with C a conjugacy class of elements of order 30 with dimension of fix space equal 8. One can check that for an orbit representative x in such sets the dimension (F ixt ∩ F ixx ) = 8, dimension(F ix(t∗x)3 ∩ F ixx ) = 12 and dimension(F ix(t∗x)6 ∩ F ixx ) = 16. Information on these orbits can be found in the following table: (t∗x)1 5 class 2A 2B (t∗x)10 class 3B 3D Dim F ix(t∗x)5 24 16 Dim F ix(t∗x)2 8 12 |UO2 (x)| ∆i Orbit Size 1 1 425590164786511872 1063975411966279680 ∆2 (t) ∆3 (t) Chapter 8 Conclusions and Future Work We end this thesis by providing a brief summary of our work and presenting a view as to potential future related research for the interested reader. 8.1 Conclusion In conclusion, we mention that this thesis is centred on analyzing the commuting involution graph of finite group G, such that G belongs to one of the following groups: i- The double cover of the symmetric group Sn ; ii- The double cover and the automorphism group of the double cover of some sporadic groups; iii- Some exceptional groups of Lie-type. Many results have been achieved throughout this work. For instance, the connectivity and disc structure of such graphs were determined. However, establishing the diameters of these graphs is the most prominent of what has been accomplished. A variety of methods have been employed for this purpose along the lines of the traditional theoretical style as is the case in the Chapter 3. On the other hand, in the remaining chapters computational approaches were most applicable in studying the commuting involution graphs. 155 8.2. Potential for Future Work 8.2 156 Potential for Future Work The results that have been obtained in this thesis open the door to examine several interesting trends in future work: 1. The completion of the study of the commuting involution graphs for the groups E7 (2) and 2 E6 (2). In addition, investigating the commuting involution graphs for more complicated groups like E8 (2). 2. Calculating the collapsed adjacency matrix for C(G, X), which is defined as follows: Let t ∈ X, set Or1 , Or2 ...Orn be a complete list of CG (t)-orbits of X and for i = 1, 2, .., n pick a representative ti ∈ Ori . Then the n × n matrix with (i, j)th entry the value |Orj ∪ ∆1 (xi )| is called the collapsed adjacency matrix for C(G, X). 3. The π-product graph Pπ (G, X) is the graph with vertex set X a non-empty subset of a finite group G with two distinct vertices x, y ∈ X connected by an edges if the product xy has order in π (a subset of the natural numbers). Valuable information about these graphs can be found in [70]. Let X be a conjugacy class of involution. If π = {2}, the P2 (G, X) is a commuting involution graph and if π is the set of odd natural numbers then Pπ (G, X) is the local fusion graph, F(G, X). This graph was studied deeply in ([14, 15]). In the future we offer the reader to study the π-product graph Pπ (G, X) in the case that X is a conjugacy class of involution, π is the set of all even natural numbers greater than 2 and G is one of the groups mentioned in this thesis. Appendix A Magma Implementations The following Magma implementations were employed to study the commuting involution graphs. However, various functions have been created for this purpose. The first one is called Orbits and aims to calculate the discs structure of the graph and the second one computes the structure constants and is denoted by StructureConstants. The last one called XCE which aims to find a random element of the set XC . r,t:=RandomElementOfOrder(G,2); C:=CentralizerOfInvolution(G,t); B:=Conjugates(G,t); repeat; z:=Random(B); until z ne t; Reps:={z}; D:=B diff {z,t}; Conj:=Conjugates(C,z); time for x in D do if x notin Conj then Conj:=Conj join Conjugates(C,x); Reps:= Reps join {x}; 157 158 end if;end for; function Orbits(S1,S2); S:={}; for x in S1 do for y in S2 do if Order(x*y) eq 2 then Include(~S,x); end if;end for;end for; return S; end function; RDisc1:=Orbits(Reps,{t}); Disc1:={};for x in RDisc1 do Disc1:=Disc1 join Conjugates(C,x);end for; R1:=Reps diff RDisc1; if #R1 ne 0 then RDisc2:=Orbits(R1,Disc1);end if; Disc2:={};for x in RDisc2 do Disc2:=Disc2 join Conjugates(C,x);end for; R2:=R1 diff RDisc2; if #R2 ne 0 then RDisc3:=Orbits(R2,Disc2);end if; Disc3:={};for x in RDisc3 do Disc3:=Disc3 join Conjugates(C,x);end for; CL:=Classes(G); CT:=CharacterTable(G); function StructureConstants(x,y) D:=0; for i:=1 to #CT do D:=D+(CT[i,x]*CT[i,y]*Conjugate(CT[i,y]))/(CT[i,1]); end for; Appendix A. Magma Implementations return (D*CL[x,2])/ #Centralizer(G,CL[y,3]); end function; function XCE(t,m); O:={};for x in Divisors(#G) do if not IsDivisibleBy(x, m) then Include(~O,x);end if;end for; s:=Id(G); repeat; x:=t^Random(G); o:=Order(t*x); if o notin O then s:= IntegerRing()!(o/m); end if; until Order((t*x)^s) eq m; return (t*(t*x)^s); end function; 159 Appendix B Magma Codes Here we provide the most used Magma codes in our work, giving their functions as described in the online handbook [32] Magma Code Random ApproximateStabilizer Description A randomly chosen element for the group G A is image of representation of G and A acts on U , a subspace or vector. Approximate the Stabilizer of U under A. CentralizerOfInvolution Given an involution g in G, this function returns the Centralizer C of g in G using an algorithm of John Bray. CharacterTable Construct the table of irreducible characters for the group G. Eigenspace The eigenspace of the matrix a, corresponding to the eigenvalue e. IsConjugate Given a group G and elements g and h belonging to G, return the value true if g and h are conjugate in G. Given a matrix group G defined over a finite field, the intrinsic returns the Centralizer in the matrix group G of g ∈ G. Given a matrix group G defined over a finite field, the intrinsic returns the conjugacy classes of G. Given a matrix group G defined over a finite field, this intrinsic returns the factored order of G. LMGCentralizer LMGConjugacyClasses LMGFactoredOrder 160 Appendix B. Magma Codes 161 Magma Code LMGIsIn Description Given a matrix group G defined over a finite field Fq , and an element x of the generic over group GL(n, q) of G; if x ∈ G then the intrinsic returns true and the corresponding element of WordGroup(G); and false otherwise. LMGRadicalQuotient Given a matrix group G defined over a finite field, the intrinsic returns a permutation group P isomorphic to G/L, where L is the soluble radical of G. An epimorphism G to P and its kernel L are also returned. LMGSolubleRadical Given a matrix group G defined over a finite field, the intrinsic returns the soluble radical S of G. A group P of type GrpPC and an isomorphism S −→ P are also returned. Given a matrix group G with natural module M and an object y which is either a vector of M , a submodule of M , or a tuple whose components are either vectors or submodules, return true if the orbit of y under G has length less than or equal to b. Otherwise the function returns false. Given a group G and some subgroup H of G, construct the ordinary character of G afforded by the permutation representation of G given by the action of G on the coset space of the subgroup H in G. A randomly chosen element for the group G Given a finite matrix group G, this intrinsic attempts to locate an element x of order n in G by random search Given a group G and a prime p, construct a Sylow psubgroup of G. 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Notations and Symbols Symbols C( −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 F ix(−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Sn+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Sn− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 XC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 [− − · · · −] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ∆i (−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Γ(−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Λ(−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 C(−, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Gx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Diam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 ω(−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ρ(−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 d(−, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 f ix(−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 supp(−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 169 Notations and Symbols 170 αi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Υ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 ς . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Qi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Uαi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 xαi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Xα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Log In

On Commuting Involution Graphs of Certain Finite Groups

On Commuting Involution Graphs of Certain Finite Groups

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