On the Equality of Certain Subgroups of the Automorphism Groups of Finite p-Groups

Abstract

Let G be a finite non-Abelian p-group, where p is a prime. An automorphism α of G is called an IA-automorphism if x−1α(x) ∈ G′ for all x ∈ G. An automorphism α of G is called an absolute central automorphism if, for all x ∈ G, x−1α(x) ∈ L(G), where L(G) is the absolute center of G. Let CIA(G)(Z(G)) and CVar(G)(Z(G)) denote, respectively, the group of all IA-automorphisms and the group of all absolute central automorphisms of G fixing the center Z(G) of G elementwise. We give necessary and sufficient conditions on a finite p-group G under which CIA(G)(Z(G)) = CVar(G)(Z(G)).