On the autocentralizer subgroups of finite p–groups

Abstract

Let G be a finite group and Aut(G) be the group of automorphisms of G. Then, the autocentralizer of an automorphism α ∈ Aut(G) in G is defined as CG(α) = {g ∈ G|α(g) = g}. Let Acent(G) = {CG(α)|α ∈ Aut(G)}. If |Acent(G)| = n, then G is an n–autocentralizer group. In this paper, we classify all n–autocentralizer abelian groups for n = 6, 7 and 8. We also obtain a lower bound on the number of autocentralizer subgroups for p–groups, where p is a prime number. We show that if p ̸= 2, there is no n–autocentralizer p–group for n = 6, 7. Moreover, if p = 2, then there is no 6–autocentralizer p–group.