Commuting Involution Graphs for Certain Exceptional Groups of Lie Type

Graphs and Combinatorics (2021) 37:1345–1355 https://doi.org/10.1007/s00373-021-02321-w (0123456789().,-volV) (0123456789().,-volV) ORIGINAL PAPER Commuting Involution Graphs for Certain Exceptional Groups of Lie Type Ali Aubad1 • Peter Rowley2 Received: 3 September 2020 / Revised: 31 March 2021 / Accepted: 12 April 2021 / Published online: 16 May 2021 Ó The Author(s) 2021 Abstract Suppose that G is a finite group and X is a G-conjugacy classes of involutions. The commuting involution graph CðG; XÞ is the graph whose vertex set is X with x; y 2 X being joined if x 6¼ y and xy ¼ yx. Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs. Keywords Commuting involution graphs  Exceptional groups of Lie type  Disc structure 1 Introduction Suppose that G is a finite group and X is a subset of G. The commuting graph, CðG; XÞ, has X as its vertex set and two vertices x; y 2 X are joined by an edge if x 6¼ y and x and y commute. The extensive bibliography in [9] points towards the many varied commuting graphs which have been studied. But here we shall be considering commuting involution graphs—these are commuting graphs CðG; XÞ where X is a G-conjugacy class of involutions. From now on X is assumed to be a Gconjugacy class of involutions. Because involutions are often centre stage in the study of non-abelian simple groups, there is a large literature on their commuting involution graphs. Indeed, such graphs have been instrumental in the construction of some of the sporadic simple groups. For example, the three Fischer groups with the conjugacy class being the 3-transpositions were investigated by Fischer [11], resulting in the construction of these groups. Later, also prior to their construction, commuting involution graphs for the Baby Monster (f3; 4g-transpositions) and the & Peter Rowley peter.j.rowley@manchester.ac.uk Ali Aubad ali.abd@sc.uobaghdad.edu.iq 1 University of Baghdad, Baghdad, Iraq 2 University of Manchester, Manchester, UK 123 1346 Graphs and Combinatorics (2021) 37:1345–1355 Monster (6-transpositions) were analyzed. Recently the commuting involution graphs of the sporadic simple groups have received much attention, see [5, 12, 14, 15, 17]. For those simple groups of Lie type consult [1, 4, 8–10], while an analysis of the commuting involution graphs of finite Coxeter groups may be found in [2, 3]. The aim of this short note is to describe certain features of CðG; XÞ when G is one of the exceptional Lie type groups of characteristic two. Specifically we consider G being one of the simple groups 3 D4 ð2Þ; E6 ð2Þ;2 F4 ð2Þ0 and F4 ð2Þ. For x 2 X we define the ith disc of x, Di ðxÞ, (i 2 N) to be Di ðxÞ ¼ fy 2 X j dðx; yÞ ¼ ig where dð; Þ is the usual distance metric on the graph CðG; XÞ. Of course, G acting by conjugation on X embeds G in the group of graph automorphisms of CðG; XÞ and, evidentily, G is transitive on the vertices of CðG; XÞ. We now choose t 2 X to be a fixed vertex of CðG; XÞ—our main focus is the description of the discs of t in CðG; XÞ. The diameter of CðG; XÞ will be denoted by Diam CðG; XÞand we shall rely upon the ATLAS [7] for the names of conjugacy classes of G. Our main result is as follows. Theorem 1 Let G be isomorphic to one of 3 D4 ð2Þ; E6 ð2Þ;2 F4 ð2Þ0 and F4 ð2Þ. The sizes of the discs Di ðtÞ are listed in Table 1 and the G-conjugacy classes of tx for x 2 Di ðtÞ; i 2 N are given in Table 2. If ðG; XÞ ¼ ðE6 ð2Þ; 2AÞ; ðE6 ð2Þ; 2BÞ; ð2 F4 ð2Þ0 ; 2AÞ; ðF4 ð2Þ; 2AÞ; ðF4 ð2Þ; 2BÞ or ðF4 ð2Þ; 2CÞ, then Diam CðG; XÞ= 2. If ðG; XÞ ¼ ð3 D4 ð2Þ; 2AÞ; ð3 D4 ð2Þ; 2BÞ; ðE6 ð2Þ; 2CÞ; ð2 F4 ð2Þ0 ; 2BÞ or ðF4 ð2Þ; 2DÞ, then Diam CðG; XÞ= 3. (i) (ii) (iii) Table 1 Disc sizes for CðG; XÞ; G ffi 3 D4 ð2Þ; E6 ð2Þ;2 F4 ð2Þ0 ; F4 ð2Þ G X ¼ tG jD1 ðtÞj jD2 ðtÞj 3 2A 18 288 512 2B 339 11112 57344 2A 127782 4954112 2B 285311 8819313408 2C 3384671 609992912640 2A 90 1664 2B 147 7712 2A 2286 67328 2B 2286 67328 2C 20944 4364800 2D 50511 113896448 D4 ð2Þ E6 ð2Þ 2 0 F4 ð2Þ F4 ð2Þ 123 jD3 ðtÞj 977994252288 3840 236912640 Graphs and Combinatorics (2021) 37:1345–1355 1347 Table 2 The conjugacy class of products tx for x 2 Di ðtÞ G X ¼ tG D1 ðtÞ D2 ðtÞ D3 ðtÞ 3 2A 2A 4A 3A 2B 2AB 3A, 4AC, 6B, 8AB D4 ð2Þ 3B, 6A, 7AD, 9AC, 12A, 13AC, 14AC, 18AC, 21AC, 28AC E6 ð2Þ 2A 2AB 3A, 4B 2B 2AC 3AB, 4AF, 4JK, 5A, 6A, 2C 2AC 3AC, 4AK,6AI, 8AJ, 12A, 5A, 7CD, 9AB, 10AB, 12B (1231205762 , 5284823044 ), 12B (5284823042 ), 12E (2818572288), 12CD, 12E (42278584323 ), 12F (8455716864), 13A, 14GH, 12F (4227858432), 12GM, 12P, 15CD, 17AB, 18AB, 20AB, 21GH, 16A, 16C, 24A 24BD, 28KL, 30EF 6D, 6F, 8C,12B 2 0 F4 ð2Þ F4 ð2Þ 2A 2AB 2B 2AB 4C, 5A 3A, 4AC, 6A, 8CD, 12AB 2A 2A, 2C 3A, 4C 2B 2B, 2C 3A, 4D 2C 2AD 3AB, 4AD, 4F, 4JM, 5A, 6GH 2D 2AD 3AC, 4AO, 6AK, 8AF, 8HK, 5A, 7AB, 8G, 9AB, 10AC, 12AB (2949124 ), 12CH, 12AB (1179648), 12IJ (1179648), 12 12IJ (294912 ), 12KO 5A, 13AB 13A,14AB, 15AB, 16AB, 17AB, 18AB, 20AB, 21AB, 24AD, 28AB, 30AB These results were obtained computationally with the aid of MAGMA [6] , GAP [16] and the ONLINE ATLAS [18]. In the course of these calculations we determined the CG ðtÞ-orbits on X (where CG ðtÞ is acting by conjugation). Representatives, in MAGMA format, for each of these orbits are to be found as downloadable files at [13], as they may be of value in other investigations of these groups. In Sect. 2 we also collate information on the action of CG ðtÞ on X. In particular, we give the CG ðtÞorbit sizes on each (non-empty) XC , XC being defined below. We observe that some ‘‘obvious’’ groups are missing in this paper. First G2 ð2Þ0 being isomorphic to PSU3 ð3Þ means it is covered in [8]. As for G ffi 2 E6 ð2Þ, the cases X ¼ 2A and X ¼ 2B are done in [1], while there are partial results in the case X ¼ 2C. Likewise [1] also has partial results for E7 ð2Þ. While E8 ð2Þ is far and away beyond current computational capabilities. We remark on the graphs studied here. First we note that as the outer automorphism of F4 ð2Þ interchanges the two classes 2A and 2B, we have that CðF4 ð2Þ; 2AÞ and CðF4 ð2Þ; 2BÞ are isomorphic graphs. A very noteworthy consequence of the present work is that the distance between t and x in CðG; XÞ is almost always determined by the G-class to which tx belongs. The exceptions are G ffi 123 1348 Graphs and Combinatorics (2021) 37:1345–1355 E6 ð2Þ; X ¼ 2C with tx 2 12B [ 12E [ 12F and G ffi F4 ð2Þ; X ¼ 2D and tx 2 12A [ 12B [ 12I [ 12J: See Table 2 for more details—for example when G ffi F4 ð2Þ; X ¼ 2D and tx 2 12I [ 12J each of 12I and 12J breaks into thirteen CG ðtÞ-orbits, 12 of size 294,912 and one of size 1,179,648 with those of size 294,912 being in D2 ðtÞ and the one of size 1,179,648 in D3 ðtÞ. A word or two about the information in our tables is required. As mentioned we employ the class names given in the ATLAS though we make some modifications. First we suppress the ‘‘slave’’ notation. So, for example, the classes 7B  2; 7C  4 of 3 D4 ð2Þ are just written as 7B, 7C, respectively. Secondly we compress the letter part of a class name when we mean the union of these classes and their letters are in alphabetical sequence. As an example, in Table 2, for G ffi F4 ð2Þ and X ¼ 2D, 8AF is short-hand for 8A [ 8B [ 8C [ 8D [ 8E [ 8F. Let C be a G-conjugacy class and define XC ¼ fx 2 X j tx 2 Cg: It is clear that XC will either be empty or be a union of certain CG ðtÞ-orbits of X (where G acts upon X by conjugation). In locating which discs of t contain the vertices in XC we sometimes need to determine how XC breaks into CG ðtÞ-orbits. Also of interest to us is the size of XC which leads us to class structure constants. Class structure constants are the sizes of sets fðg1 ; g2 Þ 2 C1  C2 j g1 g2 ¼ gg where C1 ; C2 ; C3 are G-conjugacy classes and g is a fixed element of C3 . Now these constants can be calculated directly from the complex character table of G which are recorded in the ATLAS and are available electronically in the standard libraries of the computer algebra package GAP [16]. If we take C1 ¼ C, C2 ¼ X ¼ C3 and g ¼ t, then in this case jXC j ¼ k X jGj vr ðhÞvr ðtÞvr ðtÞ ; jCG ðtÞjjCG ðhÞj r¼1 vr ð1Þ where h is a representative from C and v1 ; . . .; vk the complex irreducible characters of G. 2 CG ðtÞ-Orbits on X As promised, we tabulate the sizes of the CG ðtÞ-orbits in their action upon XC where C is a G-conjugacy class for which XC is non-empty. In the ensuing tables we use an exponential notation to indicate the multiplicity of a particular size. Thus in the table for G ffi 3 D4 ð2Þ with X ¼ 2B the entry 46 ; 2412 next to 2B is telling us that X2B is the union of eighteen CG ðtÞ-orbits, six of which have size 4 and twelve of which have size 24. Still looking at the same table, the entry 512, 1536 next to 9AC indicates that each of X9A ; X9B and X9C is the union of two CG ðtÞ-orbits of sizes 512 and 1536. We give details of the permutation ranks in Table 3. 123 Graphs and Combinatorics (2021) 37:1345–1355 1349 2.1 G ffi 3 D4 ð2Þ X ¼ 2A 2A 18 3A 512 4A 288 X ¼ 2B 2A 3, 24 2B 46 ; 2412 3A 384 3B 512 4A 5 24 ; 192 4B 2410 ; 192 4C 3846 6A 1536 6B 3846 7AC 512 7D 3072 8B 384 8 9AC 512, 1536 12A 14AC 1536 18AC 15362 21AC 8A 3846 1536 13AC 3072 3072 28AC 15362 2 2.2 G ffi E6 ð2Þ X ¼ 2A 2A 2790 2B 124992 3A 2097152 4B 2856960 X ¼ 2B 2A 63, 21602 2B 56; 4320; 302402 ; 2C 604802 ; 7257602 ; 4A 774144 4D 78643202 ; 8847360 2 967680 30720 ; 64512; 120960 3A 2359296 3B 16777216 4B 725760; 9676802 ; 4C 19353604 ; 38707204 ; 4 2 2 4423680 ; 7741440 2211840 4E 464486402 4F 20643842 ; 619315204 4J 1238630402 4K 743178240 5A 939524096 6A 707788802 6D 990904320 6F 1056964608 8C 9909043202 12B 1132462080 2 123 2A 1350 123 X ¼ 2C 3, 84, 1536, 2016 2B 2C 168, 224, 2016, 5376, 2 2 368644 ; 645124 ; 860164 8064 ; 10752 ; 16128; 2 1290243 ; 250483 ; 1032192 32256 ; 43008; 86016 3A 917504, 1572864 3B 29360128 4A 1536, 21504, 32256 4B 2 3 3 4 1536 ; 16128; 32256 3 645124 ; 1290246 ; 2580482 51609612 ; 6881282 ; 10321926 2 20643848 ; 41287684 86016; 129024 ; 258048 786432; 1032192 1032192, 1376256, 134217728 4C 36864; 43008; 64512 786432, 1032192 4D 3C 4 36864 ; 64512 ; 86016 4E 27525122 ; 11010048; 3 2580484 ; 51609610 ; 4F 12 27525122 ; 41287688 ; 22 2 55050242 ; 165150728 ; 4128768 ; 8257536 330301446 33030144 4G 41287682 ; 82575362 ; 4H 37487362 ; 660602882 4I 330301446 ; 660602887 ; 2 88080384, 264241152 66060288 13762562 ; 20643842 , 110100482 ; 165150722 ; 4K 4128768; 82575366 ; 5A 234881024, 1409286144 402653184 6C 528482304, 704643072 377487362 ; 660602882 ; 6F 88080384, 352321536 41287686 ; 825753616 ; 1651507213 ; 3303014412 ; 1651507210 ; 3303014422 660602888 ; 264241152 10 66060288 6A 2752512; 330301443 ; 6B 440401922 6D 1835008; 660602884 ; 880803843 ; 1321205764 ; 176160768, 264241152, 528482304 6E 880803842 ; 1321205762 ; 2 528482304 528482304, 704643072 1056964608 Graphs and Combinatorics (2021) 37:1345–1355 165150724 ; 330301443 ; 4J 10321922 ; 20643842 ; 10 1032192 ; 2064384 ; 16515072 962 ; 5376; 161283 ; 322564 6G 2818572288 6H 10569646082 ; 4227858432 6I 8455716864 7C 805306368 7D 3221225472 8A 15728642 ; 330301442 ; 377487362 ; 660602882 ; 880803842 ; 1321205762 5284823042 8B 2 2 2 2 8C 37748736 ; 44040192 ; 8 2 8D 16515072 ; 33030144 ; 1321205762 ; 26424115220 10569646082 13212057612 ; 26424115210 1321205762 ; 5284823042 2 2 66060288 ; 88080384 ; 66060288 ; 88080384 ; 8E 2 4 176160768 ; 528482304 ; 8F 21139292165 8G 264411524 ; 5284823044 ; 10566460812 2 2113929216 3 6 10569646082 ; 21139292166 ; 8H 2113929216 8I 427858432 8J 9A 22548578304 9B 3221225472, 9663676416 10A 28185722882 ; 4227858432 Graphs and Combinatorics (2021) 37:1345–1355 continued 42278584324 10B 2 2818572288; 4227858432 2 12A 402653184 12B 1321205762 ; 5284823046 12D 14092861442 ; 21139292164 ; 12E 2818572288; 42278584323 12H 3523215362 ; 21139292166 8455716864, 16911433728 12C 2642411528 ; 5284823044 12F 4227858432, 8455716864 12I 2 105696460816 4227858432 12G 5637144576 42278584324 8455716864 12J 2 6 1056964608 ; 2113929216 ; 12K 14092861442 ; 42278584324 84557168644 8 4227858432 16911433728 12M 169114337282 12P 169114337282 13A 19327352832 14G 16911433728 14H 9663676416 15C 22548578304 15D 7516192768, 22548578304 16A 84557168644 1351 123 12L 1352 123 continued 16C 169114337284 17A 45097156608 17B 45097156608 18A 96636764162 18B 67645734912 20A 169114337282 20B 338228674564 21G 19327352832 8 21H 45097156608 16911433728 24C 338228674562 2 28L 33822867456 4 24B 24D 2 33822867456 28K 9663676416 30E 225485783042 30F 67645734912 24A 8455716864 Graphs and Combinatorics (2021) 37:1345–1355 Graphs and Combinatorics (2021) 37:1345–1355 Table 3 Class sizes and permutation rank 1353 G X ¼ tG |X| 3 2A 819 2B 68796 2A 5081895 2B 8822169720 2C 1587990549600 2A 1755 5 2B 11700 30 2A 69615 5 2B 69615 2C 4385745 2D 350859600 D4 ð2Þ E6 ð2Þ 2 F4 ð2Þ0 F4 ð2Þ Permutation rank 4 27 5 62 719 5 33 1002 2.3 G ffi 2 F4 ð2Þ0 X ¼ 2A 2A 10 2B 80 4C 640 5A 1024 2A 3, 12 2B 123 ; 482 3A 2562 4A 1922 4B 962 4C 96; 1922 5A 768 6A 7682 8CD 3842 12AB 7682 13AB 1536 X ¼ 2B 2.4 G ffi F4 ð2Þ X ¼ 2A 2A 270 2C 2016 3A 32768 4C 34560 270 2C 2016 3A 32768 4D 34560 X ¼ 2B 2B 123 1354 Graphs and Combinatorics (2021) 37:1345–1355 X ¼ 2C 2AB 30 2C 322 ; 180; 19202 2D 7202 ; 9604 ; 11520 3AB 32768 4AB 15360 4CD 11520 4F 10242 4JK 307202 4L 737280 4M 1843202 5A 1048576 6GH 983040 9; 122 ; 242 ; 724 ; 2D X ¼ 2D 244 ; 14429 ; 57624 2AB 3; 12; 722 ; 192 2C 3AB 2048, 6144, 24576 3C 262144 4AB 192; 5768 ; 11524 9216, 12288 4CD 1444 ; 1923 ; 2884 4EF 5764 ; 15364 4GH 23044 ; 46086 ; 92162 1447 ; 1922 ; 5764 115216 ; 9216 23044 ; 92168 57613 ; 11522 ; 23044 ; 184324 ; 73728 4 4608 ; 12288 4I 921614 ; 184328 4 4JK 2 20 36864 ; 73728 4M 4L 16 92169 ; 368648 1474562 22 4608 ; 9216 ; 18432 23042 ; 460812 ; 921630 36 11524 ; 15364 ; 23044 4N 1474564 4O 3686412 ; 1474564 6AB 61442 ; 245762 ; 737283 6CD 368642 ; 491522 2 18432 ; 36864 5A 1966082 ; 589824 737283 ; 294912 6EF 786432 6GH 6K 2359296 7AB 8B 8 2 2 12288; 36864 ; 49152 6IJ 737288 ; 1474564 ; 2949124 8A 2949124 8G 5898242 737287 ; 1474562 ; 294912 4 147456 ; 294912 8CF 1572864 2 10 24576 ; 73728 4 4 147456 ; 294912 8HI 2949126 8J 58982416 8K 5898246 9AB 1572864, 4718592 10AB 5898242 ; 11796482 10C 5898242 ; 11796484 12EH 983042 ; 2949124 ; 5898244 12MN 58982414 12AB 4 294912 ; 1179648 12 12IJ 294912 ; 1179648 4 12CD 12KL 2 786432 2 2359296 12O 2359296 13A 9437184 14AB 4718592 15AB 1572864, 4718592 16AB 23592964 17AB 9437184 18AB 47185922 20AB 23592964 21AB 9437184 24AD 4 28AB 2 30AB 47185922 2359296 4718592 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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