Profinite groups with few conjugacy classes of $p$-elements

TL;DR

This paper characterizes profinite groups with fewer than continuum many conjugacy classes of p-elements, showing that for odd primes this occurs precisely when the p-Sylow subgroups are finite, with a weaker result for p=2.

Contribution

It establishes a clear criterion linking the number of conjugacy classes of p-elements to the finiteness of p-Sylow subgroups in profinite groups.

Findings

Fewer than continuum conjugacy classes of p-elements imply finite p-Sylow subgroups for odd p.

A similar but weaker result holds for p=2.

Provides a characterization of the structure of profinite groups based on conjugacy class count.

Abstract

It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either has $p$-power order or topologically generates a group isomorphic to ${\mathbb Z}_p$.) A weaker result is proved for $p=2$.