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 α of G is called an IA n -automorphism if x− 1 α(x) ∈ γn +1(G) for all x ∈ G. The group of all IA n -automorphisms of G is denoted by IA n (G). Let denote the subgroup of IA n (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 , 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 Aut z (G) = IA n (G), and obtain Theorem B of Attar [4] as a particular case.