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

Mathematics Subject Classification (2020):

• 20D45
• 20D15

Key words:

• IAⁿ-automorphism
• central automorphism
• absolute central automorphism