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 γ_n (G) and Z_n (G) respectively denote the nth term of the lower and upper central series of G, and let L_n (G) denote the nth absolute center of G. An automorphism α of G is called an IA ⁿ -automorphism if x^{− 1} α(x) ∈ γ_{n +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 Z_n (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 γ_{n +1}(G) ≤ M ≤ Z(G) ∩ L_n (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.