Profinite groups with few conjugacy classes of 𝑝-elements

Wilson, John

doi:10.1090/proc/15925

Profinite groups with few conjugacy classes of $p$-elements
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Proc. Amer. Math. Soc. 150 (2022), 3297-3305 Request permission

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 $p$-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 $p=2$.

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