Profinite groups in which centralizers are abelian

TL;DR

This paper studies profinite groups where all centralizers are abelian, showing such groups are virtually pronilpotent with specific structural properties and detailed information on their finite quotients.

Contribution

It proves that profinite CA-groups have a normal open subgroup that is either abelian or pro-p, and provides detailed structure of their finite quotients.

Findings

Profinite CA-groups are virtually pronilpotent.

Existence of a normal open subgroup that is abelian or pro-p.

Detailed description of finite quotients G/N.

Abstract

The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, a rather detailed information about the finite quotient G/N is obtained.