Infinite Groups with an Anticentral Element

Abstract

An element of a group is called anticentral if the conjugacy class of that element is equal to the coset of the commutator subgroup containing that element. A group is called Camina group if every element outside the commutator subgroup is anticentral. In this paper, we investigate the structure of locally finite groups with an anticentral element. Moreover, we construct some non-periodic examples of Camina groups, which are not locally solvable.

Key Words:

• Anticentral elements
• Camina groups
• Locally finite groups

2010 Mathematics Subject Classification:

• 20F50
• 20E45

ACKNOWLEDGMENT

This study is carried out under the supervision of Professor Patrizia Longobardi and Professor Mercede Maj of University of Salerno. I would like to express my graditude to Professor Longobardi and Professor Maj for their suggestions and comments. I also would like to thank Professor David Chillag and Professor Marcel Herzog since they suggested to look at anticentral elements. I would like to thank University of Salerno for hospitality and the colleagues in University of Salerno for their kindness and hospitality. This study is supported by The Scientific and Technological Research Council of Turkey (TÜBTAK) BDEB 2219 International Post Doctoral Research Fellowship. I would like to thank TÜBTAK for the support.

Notes

Communicated by A. Turull.