On Autocentral Automorphisms of Finite p-Groups

Abstract

Let G be a finite group. An automorphism $$\alpha $$ of G is called an $$\text{ IA }$$ automorphism if it induces the identity automorphism on the abelianized group $$G/G'$$ . Let $$\text{ IA }(G)$$ be the group of all $$\text{ IA }$$ automorphisms of G, and let $$\text{ IA }(G)^*$$ be the group of all $$\text{ IA }$$ automorphisms of G which fix Z(G) element-wise. Let $$\mathrm {Var}(G)$$ be the group of all autocentral automorphisms of G. We find necessary and sufficient conditions for a finite p-group G such that $$\text{ IA }(G)^*=\text{ Var }(G)$$ . We also extend the famous result of Adney and Yen (Theorem 1 of Ill J Math 9:137–143, 1965).