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 $G$.
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$\begingroup$ The product of all $\mathbb{Z}/p$ for all primes $p$ is a profinite group with infinite torsion and yet satisfies your hypothesis. I don't know if you'll be able to find a useful general criterion, though you may be interested in the book by Ribes and Zalesskii, particularly their chapter 4.7 on "Torsion in the profinite completion of a group". $\endgroup$– Will ChenCommented Sep 8, 2018 at 16:35
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1$\begingroup$ A profinite abelian group is a product over primes $p$ of pro-$p$-groups. So your groups are just products over $p$ of abelian pro-$p$-groups with finite torsion, that is, extension with finite kernel of $\mathbf{Z}_p^{X_p}$ for some set $X_p$. $\endgroup$– YCorCommented Sep 8, 2018 at 22:48
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