Profinite groups in which centralizers are virtually procyclic

TL;DR

This paper characterizes profinite groups with specific centralizer structures, showing they are either virtually pro-p or virtually torsion-free procyclic, with implications for their algebraic structure and rank.

Contribution

It provides a classification of profinite groups based on the nature of their centralizers, extending understanding of their structural properties.

Findings

G is either virtually pro-p or virtually torsion-free procyclic

If G is not pro-p, then G has finite rank

Centralizers of elements determine the group's overall structure

Abstract

The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic. The same conclusion holds for profinite groups in which the centralizer of every nontrivial element is virtually procyclic; moreover, if G is not pro-p, then G has finite rank.