Question 1: Does there exist an intrinsic characterization of groups $G$ isomorphic to some subgroup of some finitary symmetric group (i.e. all the permutations of a given set that fix all but finitely many elements)?

Clearly every such $G$ enjoys local finiteness, but I see where (for a fixed $p$) the multiplicative group $\{e^{2\pi i k/p^n}\}$ shows that this does not suffice (because all non-identity elements have $m$-th roots for every $m$).

Question 2: Do there exist locally finite groups not isomorphic to any subgroup of any finitary symmetric group such that no non-identity element has $m$-th roots for every $m$?