Looking for an example of profinite groups

Question

Is there a profinite group $G$ with a locally finite subgroup $H$ such that $\overline H$, the closure of $H$, is not torsion?

Answer 1

You can take $G=\prod_{k>0}\mathbb{Z}/2^k$ and $H=\bigoplus_{k>0}\mathbb{Z}/2^k$. Then every finitely generated subgroup of $H$ is finite, and $\overline{H}=G$, but the element $(1,1,1,\dotsc)\in G$ is not torsion.

Answer 2

Actually, if you take an infinite, finitely generated, residually finite, torsion group, then its profinite closure is not torsion. This follows from Zelmanov's solution of the general Burnside problem for compact groups which is saying that finitely generated, torsion, profinite group is finite. Thus, for example, the profinite completion of the Grigorchuk group is not torsion.