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

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 $p=2$.

Corrected version of a paper to appear in Proc. Amer. Math. Soc. (2022), with expanded explanations