TITLE: Locally finite compact groups
QUESTION [5 upvotes]: I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it be finite? Is there any structure theorem for this kind of groups?
For the larger class of torsion compact groups structure theorems exist. In particular, They tell us that these groups are profinite. So, locally finite compact groups are profinite as well. But, can one prove more?

REPLY [4 votes]: To summarize some of what is stated in the comments above, there is a related question (posed as Question 4.8.5b on p. 156 of Ribes-Zalesskii) of whether torsion profinite groups have bounded exponent. As noted in the comments, your question is equivalent to this question due to Zelmanov's solution of the restricted Burnside problem, which implies that finitely generated profinite torsion groups are finite (see again p. 156 of Ribes-Zalesskii for discussion). In turn, the case of whether profinite torsion groups have bounded exponent may be reduced to the case of pro-$p$ torsion groups by a theorem of Wilson (4.8.5d in Ribes-Zalesskii, which suffices for the finitely generated case).