On Infinite Camina Groups

Abstract

A group G is called a Camina group if G′ ≠ G and each element x ∈ G∖G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).

Key Words:

• Camina groups
• Centralizers
• Cosets
• Periodic groups
• Residually finite groups

2000 Mathematics Subject Classification:

• Primary 20E45, 20E34
• Secondary 20F16, 20F18

ACKNOWLEDGMENTS

The first author is grateful to the Department of Mathematics and Informatics of the University of Salerno for its hospitality and support, while this investigation was carried out.

Notes

Communicated by A. Turull.