Main Article Content
On the coincidence of automorphism groups of finite p-groups
Abstract
Let G be a finite non-abelian p-group, where p is a prime. Let γn(G) and Zn(G) respectively denote the nth term of the lower and upper central series of G, and let Ln(G) denote the nth absolute center of G. An automorphism a of G is called an IAn-automorphism if x-1 a(x) ∈ γn+1 (G) for all x ∈ G. The group
of all IAn -automorphisms of G is denoted by IAn (G). Let IAnZn(G) (G) denote the subgroup of IAn (G) which fixes Zn(G) elementwise. In this paper, we give necessary and sufficient conditions for a finite non-abelian p-group G of class (n+1) such that IAnZn(G) = AutMLn(G) (G), where M is a subgroup of G such that γn+1(G) ≤ M ≤ Z(G) ⋂ Ln(G), and as a consequence, we obtain the main result of Chahal, Gumber and Kalra [7, Theorem 3.1]. We also give necessary and sufficient conditions for a finite non-abelian p-group G of class (n+1) such that Autz(G) = IAn(G), and obtain Theorem B of Attar [4] as a particular case.
