Groups with Anticentral Elements

Abstract

We study finite groups G with elements g such that |C _G (g)| = |G:G′|. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class of nilpotent supplements for the commutator subgroup and, using the classification of finite simple groups, that these groups are solvable.

Key Words:

• Carter subgroup
• Centralizers
• Finite groups
• Solvable groups

2000 Mathematics Subject Classification:

• Primary 20D10
• Secondary 20D15, 20C15

ACKNOWLEDGMENTS

This work is part of my thesis done under the supervision of Knörr. I would like to thank him for his many helpful remarks in our conversations about this article, and especially for suggesting the subject of “anticentral” elements.

Notes

Communicated by A. Turull.