On Finite n-Acentralizer Groups

Abstract

Let G be a group and Aut(G) be the group of automorphisms of G. Then the Acentralizer of an automorphism α ∈Aut(G) in G is defined as C _G (α) = {g ∈ G∣α(g) = g}. For a finite group G, let Acent(G) = {C _G (α)∣α ∈Aut(G)}. Then for any natural number n, we say that G is n-Acentralizer group if |Acent(G)| =n. We show that for any natural number n, there exists a finite n-Acentralizer group and determine the structure of finite n-Acentralizer groups for n ≤ 5.

Key Words:

• Automorphism
• Centralizer
• Finite groups

2010 Mathematics Subject Classification:

• 20D45
• 20D25

ACKNOWLEDGMENT

The authors would like to thank the referee for his/her helpful comment.

Notes

Communicated by A. Olshanskii.