CENTRAL AUTOMORPHISM GROUPS FIXING THE CENTER ON THE NILPOTENT GROUPS

By submitting this extended abstract to 6th group theory conference I confirm that (i) I and any other coauthor(s) are responsible for its content and its originality; (ii) any possible coauthors agreed to its submission to 6th group theory conference. CENTRAL AUTOMORPHISM GROUPS FIXING THE CENTER ON THE NILPOTENT GROUPS MOHAMMAD MEHDI NASRABADI1 AND ZAHRA KABOUTARI FARIMANI2∗ 1 Department of Mathematics, University of Birjand, Birjand, Iran. mnasrabadi@birjand.ac.ir 2 Department of Mathematics, University of Birjand, Birjand, Iran. kaboutarizf@gmail.com Abstract. Let G be a finite nilpotent group and Autc (G) be the ( ) group of all central automorphisms of G. Let C ∗ (G) = CAutc (G) Z(G) be the group of all central automorphisms of G fixing Z(G) elementwise. In this paper we give conditions on G such that Autc (G) = C ∗ (G). Also we mention to finitely generated p-group G such that each central automorphism of G fixes the center of G elementwise. 1. Introduction In this paper p denotes a prime number. Let G be a group. We denote by G′ , Z(G) and Aut(G), respectively the commutator subgroup, the center and the automorphism group of G. For a group G and an abelian group H, Hom(G, H) denotes the group of all homomorphisms from G to H. An automorphism α of G is called a central automorphism if x−1 α(x) ∈ Z(G) for each x ∈ G. The central automorphisms of G, denoted by Autc (G), fix G′ elementwise and form a normal subgroup of the full automorphism group of G. We denote by C ∗ (G) the 2010 Mathematics Subject Classification. 20D15, 20D45, 20E36. Key words and phrases. central automorphisms, nilpotent groups, finitely generated groups. ∗ Speaker. 1 2 NASRABADI AND KABOUTARI group of all central automorphisms of G fixing Z(G) elementwise. Yadav in [5] obtained some necessary and sufficient conditions for the equality Autc (G) = C ∗ (G) on p-groups of class 2. Then Jafari in [2] and Attar in [4] characterized finite p-groups G of arbitrary nilpotency class such that Autc (G) = C ∗ (G). We research this equality on finite nilpotent groups. Also in the end we mention to finitely generated p-group G in the case Autc (G) = C ∗ (G). 2. Preliminary results First we give some results that are needed for the main results of this paper. Definition 2.1. A non-abelian group G is called purely non-abelian if it has no nontrivial abelian direct factor. Proposition 2.2. Let G be a group. Then C ∗ (G) ∼ = Hom( G , Z(G)). Z(G) G Proof. Consider the map fσ : Z(G) −→ G defined by fσ (gZ(G)) = −1 g σ(g) for all g ∈ G and each σ ∈ C ∗ (G). Since every element of C ∗ (G) fixes each element of Z(G), fσ is well known. Now it is easy to G check θ : C ∗ (G) −→ Hom( Z(G) , Z(G)) defined by θ(σ) = fσ , for any ∗ σ ∈ C (G), is an isomorphism. □ Theorem 2.3. ([2]) Let G be a finite p-group. Then Autc (G) = C ∗ (G) n if and only if Z(G)G′ ⊆ Gp G′ where exp(Z(G)) = pn . Lemma 2.4. ([5, Lemma 2.4]) Let G be a finite p-group such that Autc (G) = C ∗ (G). Then G is purely non-abelian. Lemma 2.5. ([1, Lemma 2.3] Let G be a finitely generated nilpoG = ⟨x1 Z(G)⟩ × · · · × ⟨xd Z(G)⟩ for some tent group of class 2 and Z(G) x1 , . . . , xd ∈ G. Then (i) G′ = ⟨[xi , xj ] : 1 ≤ i < j ≤ d⟩. G G (ii) If Z(G) is torsion, then G′ is torsion and exp( Z(G) ) = exp(G′ ). Lemma 2.6. ([3, 5.2.22]) Let G be a nilpotent group. If G is finitely generated and infinite, then Z(G) contains an element of infinite order. 3. Main Results Yadav in [5] find necessary and sufficient conditions on a finite pgroup G of class 2 such that Autc (G) = C ∗ (G). In this section we mention to finitely generated p-group G. Let G be a group. We start with the results on nilpotent groups that Autc (G) = C ∗ (G). CENTRAL AUTOMORPHISMS GROUPS FIXING THE CENTER... 3 Lemma 3.1. Let G be a finite group such that G = H × K where H and K are normal subgroups of G. If (|H|, |K|) = 1 then Autc (G) ∼ = Autc (H) × Autc (K). By the results in [4] and [5] we have Lemma 3.2. Let G be a finite nilpotent group such that G = P1 ×· · ·× Pn where Pi ’s are distinct sylow subgroups of G for 1 ≤ i ≤ n. Then C ∗ (G) ∼ = C ∗ (P1 ) × · · · × C ∗ (Pn ). Corollary 3.3. Let G be a finite nilpotent group and {P1 , . . . , Pn } be set all distinct sylow subgroups of G. Then the following are equivalent. (a) Autc (G) = C ∗ (G). (b) Autc (Pi ) = C ∗ (Pi ) for 1 ≤ i ≤ n. (c) Z(Pi )Pi ′ ⊆ Pi s Pi ′ where exp(Z(Pi )) = s. We know if G is abelian, then Autc (G) = Aut(G) and the only automorphism fixing Z(G) = G is the identity map. Thus C ∗ (G) = ⟨1⟩. Lemma 3.4. Let G be an abelian finitely generated group. Then Autc (G) = C ∗ (G) if and only if G ≃ C2 , where C2 is the cyclic group of order 2. Proof. Let Autc (G) = C ∗ (G). Since G is abelian group, C ∗ (G) = ⟨1⟩. Thus Autc (G) = ⟨1⟩ and Aut(G) = ⟨1⟩, hence G ≃ C2 . Conversly, it is easy to check the result is true. □ Therefore now consider non-abelian groups. Lemma 3.5. Let G be a finitely generated non-abelian p-group of class 2. If Autc (G) = C ∗ (G), then G is a purely non-abelian group. Proof. Assume contrarily that G is not purely non-abelian. Then G = H × K where H is purely non-abelian and K is nontrivial abelian subgroup of G. Each subgroup of a finitely generated nilpotent group is finitely generated. Thus K and H are finitely generated. We have Z(H) ̸= 1 and so Hom(K, Z(H)) is nontrivial. We can easily prove Hom(K, Z(H)) is a subgroup of Autc (G). But Hom(K, Z(H)) ⊈ C ∗ (G), which is a contradiction. □ Remark: The above lemma is true for arbitrary nilpotency class. Now we formulate the following problem. Problem 3.6. Classify all finitely generated p-groups G of class 2 such that Autc (G) = C ∗ (G). Definition 3.7. The group G is called stem group if Z(G) ⊆ G′ . Proposition 3.8. Let G be a finitely generated infinite group of class G 2 and Z(G) be finite. Then G′ ̸= Z(G). This means there is no stem group with above conditions. 4 NASRABADI AND KABOUTARI G Proof. Let G be a finitely generated infinite group of class 2 and Z(G) be finite. Then by lemma 2.5, G′ is torsion. Also since G is infinite, by lemma 2.6, Z(G) contains an element of infinte order. So G′ ̸= Z(G). Hence G can not be a stem group. □ References 1. Z. Azhdari, M. Akhavan-Malayeri, On inner automorphisms and central automorphisms of nilpotent group of class 2, J. Algebra Appl. (10) 4 (2011), no. 6, 1283–1290. 2. S. H. Jafari, Central automorphism groups fixing the center elementwise, International electronic journal of algebra. (9) (2011), 167–170. 3. D. J. S. Robinson, A course in the theory of groups, Springer-Verlag, Berlin, 1995. 4. M. Shabani Attar, Finite p-groups in which each central automorphism fixes centre elementwise, Arch. Math. 40 (2012), 1096–1102. 5. M. K. Yadav, On central automorphisms fixing the center element-wise, Comm. Algebra. 37 (2009), 4325–4331.