On the autocentralizer subgroups of finite p-groups

Let G be a finite group and Aut(G) be the group of automorphisms of G. Then, the autocentralizer of an automorphism alpha is an element of Aut(G) in G is defined as C-G(alpha) = {g is an element of G vertical bar alpha(G) = g}. Let Acent(G) = {C-G (alpha)vertical bar alpha is an element of Aut(G)}. If vertical bar Acent(G)vertical bar = 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 not equal 2, there is no n-autocentralizer p-group for n = 6,7. Moreover, if p = 2, then there is no 6-autocentralizer p-group.