  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 $p=2$.
