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Nilpotent Groups Related to an Automorphism

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Nilpotent Groups Related to an Automorphism Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:60 https://doi.org/10.1007/s12044-018-0441-0 Nilpotent groups related to an automorphism AHMAD ERFANIAN∗ and MASOUMEH GANJALI Department of Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran *Corresponding author. E-mail: [email protected]; [email protected] MS received 16 March 2017; revised 23 May 2017; accepted 18 June 2017; published online 25 October 2018 Abstract. The aim of this paper is to state some results on an α-nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2) (2015) 271–283), for any fixed automorphism α of a group G. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur’s theorem. In the last section of the paper, we study absolute normal subgroups of a finite group. Keywords. Nilpotent group; identity nilpotent group; absolute normal subgroup. 2010 Mathematics Subject Classification. Primary: 20F12; Secondary: 20D45. 1. Introduction The extension of a nilpotent group has been studied by different authors. For instance an autonilpotent group has been investigated in [13]. Recently, Barzegar and Erfanian [2] defined a new extension of nilpotent and solvable groups. Actually, this extension displays nilpotency and solvability of a group with respect to an automorphism. Similar to the definition of a nilpotent group, this extension needs an introduction of a normal series of group G. Assume that α is an automorphism of group G. Define the subgroup α − − α Z (G) ={y ∈ G :[x, y]α = x 1 y 1xy = 1 ∀x ∈ G} which is called the α- center of G. A central α-series is a normal series {1}=G0 G1 ··· Gn = G α = / ≤ α¯ ( / ) α¯ ∈ ( / ) such that Gi Gi and Gi+1 Gi Z G Gi , where Aut G Gi by the rule α¯ = α ∈ / ≤ ≤ − α gGi g Gi for all gGi G Gi ,0 i n 1. An -nilpotent group is a group which has at least a central α-series. One can easily see that Z α(G) = Z(G) ∩ Fix(α), where Fix(α) ={g ∈ G : gα = g}. As a result of Theorem 3.10 of [2], Z α(G) ={1} whenever G is α-nilpotent, so G is not α-nilpotent for a fixed point free automorphism α. Now, put α( ) α α α¯ G Zi G Z (G) = Z (G) and define Z ( α ( ) ) = α ( ) for i ∈ N. Then the normal series 1 Zi−1 G Zi−1 G { }= α( ) α( ) α( ) ··· 1 Z0 G Z1 G Z2 G is said to be an upper central α-series. One of the interesting results in [2] is a theorem that provides a sufficient and necessary condition for a given group to be α-nilpotent. It © Indian Academy of Sciences 1 60 Page 2 of 12 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:60 states that a group G is α-nilpotent if and only if there exists a positive integer n such α( ) = α that Zn G G. Also, Barzegar and Erfanian defined, in an analogous manner, the - ( ) =[ , ] : , ∈ α( ) = α( ) = ( ) commutator subgroup Dα G x y α x y G . Put 1 G G, 2 G Dα G α ( ) =[ ,α( )] =[ , ] : ∈ , ∈ α( ) and define inductively n+1 G G n G α x y α x G y n G for all n ≥ 1. It has been proved that a group G is α-nilpotent if and only if there exists an integer α( ) ={ } α r such that r G 1 . It is easy to see that if G is an -nilpotent group, H G and H α = H, then G/H is α¯ -nilpotent. Also we can prove that for every normal subgroup N contained in Z α(G),ifG/N is α¯ -nilpotent, then G is α-nilpotent. One can check that an α- nilpotent group is nilpotent, but the converse is not true in general. Because consider a group with a fixed point free automorphism α of prime order which is nilpotent (Theorem 10.5.4 α α α( ) = ( ) of [14]), but is not -nilpotent. For an inner automorphism of group G, Zn G Zn G for all n ∈ N, so nilpotency and α-nilpotency are equivalent. Therefore, one question that may arise is: can we find a non-inner automorphism α of a nilpotent group G such that G is α-nilpotent? In section 3, we answer this question for abelian groups, indeed we classify all finite groups that are α-nilpotent for at least one non-identity automorphism α. Suppose that α ∈ Aut(G) and β ∈ Aut(H), then we define α × β as an automorphism of G × H by α × β : (g, h) → (gα, hβ ) for all g ∈ G and h ∈ H. By Theorem 3.11 of [2], if G is α-nilpotent and H is β-nilpotent, then G × H is α × β-nilpotent. Also we know that a finite nilpotent group is the direct product of its Sylow subgroups, so it is enough to answer the above question for finite p-groups. We remind an old result proved by Gaschütz [7], asserting that finite non-abelian p- groups possess non-inner automorphisms of p-power order. Also, some have investigated finite non-abelian p-groups with at least one non-inner automorphism of order p (for example, see [1] and [4]). Here, we prove a theorem that states the relation between α- nilpotency of a group and the order of α. Although, the above question has not been answered as yet for non-abelian groups, but we do hope to give a certain answer by using results of this paper and papers similar to [1], [4] and [7] in near future. In section 3, we will give an example of a group that is α-nilpotent if and only if α is the identity automorphism of G. We call a group with this property an identity nilpotent group and we classify all finitely generated identity nilpotent groups. At the end of this paper, we recall the definition of absolute normal subgroups [5], and study absolute normal subgroups of some finite groups. All groups in this paper are considered finite unless we notify. We show the identity element of G and Aut(G) by 1 and I respectively. 2. Basic results In this section, we investigate some conditions that a nilpotent group G is α-nilpotent for a non-identity automorphism α. Also, we show that an α-solvable group is αn-solvable for an arbitrary positive integer n. Moreover, we are going to give a generalization of the converse of Schur’s theorem. Theorem 2.1. If G is a finite group and I = α ∈ Aut(G) such that (|G|, |α|) = 1, then G is not α-nilpotent. Proof. Assume that (|G|, |α|) = 1 and G is α-nilpotent. Then there exists a normal series, { }= ≤ ≤ ··· ≤ = α = / ≤ α¯ ( / ) 1 G0 G1 Gn G such that Gi Gi and Gi+1 Gi Z G Gi for α¯ 0 ≤ i ≤ n − 1. So for an arbitrary g ∈ Gi+1,wehavegGi ∈ Gi+1/Gi ≤ Z (G/Gi ) = Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:60 Page 3 of 12 60 (α)¯ ∩ ( / ) ∈ (α)¯ α¯ = . α Fix Z G Gi . Thus, gGi Fix and gGi gGi and by Theorem 1 6 6of[9], is identity, which is a contradiction. Hence, G is not α-nilpotent. The structure of automorphisms of an extra-special p-groupmaybeusedinthefollowing example, verified in [15]. Example 2.2. If P is an extra-special p-group of order p2n+1, then Aut(P) =θ H, where θ ∩H ={I }, |θ|=p − 1 and H is a normal subgroup of Aut(P) consisting of all members of Aut(P) which act trivially on Z(P) and p is an odd prime number. Since (|P|, |θ|) = 1, then by Theorem 2.1, P is not θ-nilpotent. α( ) [ , ] ∈ α ( ) The fact that an element x is a member of Zn G if and only if y x α Zn−1 G for all y ∈ G, may used in the rest of this paper. DEFINITION 2.3 An automorphism α of a group G is central if α commutes with every inner automorphism or equivalently if g−1gα ∈ Z(G), for all g ∈ G. Lemma 2.4.Ifα is a central automorphism that fixes Z(G) element wise, then G is α- nilpotent if and only if G is nilpotent. ∈ N ( ) = α( ) Proof. We are going to prove that for every n , Zn G Zn G . It is not difficult α( ) ≤ ( ) ( ) ≤ (α) to see that Zn G Zn G in general. By assumption in lemma, Z G Fix , then α( ) = ( ) α ( ) = ( ) ∈ ( ) ∈ Z G Z G . Now, suppose that Zn−1 G Zn−1 G and x Zn G and y G are [ , ] =[ , ] −1 α [ , ]∈ ( ) = α ( ) two arbitrary elements. Then y x α y x x x and y x Zn−1 G Zn−1 G , −1 α ∈ ( ) = α( ) ≤ α ( ) [ , ] −1 α =[ , ] ∈ α ( ) also x x Z G Z G Zn−1 G . Hence, y x x x y x α Zn−1 G ∈ ∈ α( ) ( ) ≤ α( ) α for all y G and so, x Zn G . Therefore, Zn G Zn G and -nilpotency and nilpotency are equivalent. A power automorphism of a group G is an automorphism which maps every subgroup of G into itself. Lemma 2.5.Ifα, β ∈ Aut(G), β is a power automorphism and G is α-nilpotent. Then it is αβ -nilpotent, where αβ = β ◦ α ◦ β−1.
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