Finite Groups with a Pre-Fixed-Point-Free Power Automorphism

Abstract

A power automorphism θ of a group G is said to be pre-fixed-point-free if C_G(θ) is an elementary abelian 2-group. G is called an E-group if G has a pre-fixed-point-free power automorphism. In this paper, finite E-groups, together with all their pre-fixed-point-free power automorphisms, are completely determined. Moreover, a characteristic of finite abelian groups is given, which explains some known facts concerning power automorphisms.

Key Words:

• Minimal nonabelian groups
• Power automorphisms
• p-Groups

2010 Mathematics Subject Classification:

• 20D15
• 20E34

ACKNOWLEDGMENT

The authors are very grateful to the referee who read the manuscript carefully and provided a lot of valuable suggestions and useful comments.

The research of the work was partially supported by the National Natural Science Foundation of China (11171169), the National Natural Science Foundation of China (11071155), and the Shanghai Leading Academic Discipline Project (J50101).

Notes

Communicated by J. Zhang.