Let $G$ be a profinite abelian group such that for every $x\in G$ and every $n\in\mathbb Z$ the preimage of $x$ under the multiplication by $n$ map is finite. Does it follow that the torsion subgroup of $G$ is finite? (I suppose not). If not, is there any criterion for a profinite group to have finite torsion? The answer is yes in the case when $G$ is a pro-p-group. Indeed, $l$-torsion for every $l\ne p$ is trivial and $p$-primary torsion is finite as there are no infinitely $p$-divisible elements in $G$.