Finitary symmetric group is locally finite

Statement
The fact about::finitary symmetric group on any set is a fact about::locally finite group: every finitely generated subgroup of it is finite.

In particular, the finitary symmetric group on an infinite set is not finitely generated.

Facts used

 * 1) uses::Local finiteness is directed union-closed

Proof idea
Any permutation of the finitary symmetric group moves only finitely many elements of the set. Thus. a finite set of permutations can, in total, move only finitely many elements of the set. Hence, the subgroup generated by these permutations must lie inside the symmetric group on that finite subset, which is finite.

More general proof idea
The more general idea is to use fact (1), combined with the fact that the finitary symmetric group is the directed union of the symmetric groups on all finite subsets.