Baer-Schreier-Ulam theorem

Statement
Let $$S$$ be an infinite set and $$\operatorname{Sym}(S)$$ denote the symmetric group on $$S$$. For every cardinal $$\alpha \le |S|$$, define $$\operatorname{Sym}_\alpha(S)$$ as the group of all permutations on $$S$$ that move at most $$\alpha$$ elements. Then, the normal subgroups of $$\operatorname{Sym}(S)$$ are as follows:


 * The trivial subgroup.
 * The finitary alternating group: The group of all even finitary permutations.
 * The finitary symmetric group: The group of all finitary permutations.
 * The subgroups $$\operatorname{Sym}_{\alpha}(S)$$ for all ordinals $$\alpha \le |S|$$, where $$\operatorname{Sym}_{\alpha}(S)$$ is the group of permutations whose support has size equal to the cardinality of any ordinal smaller than $$\alpha$$.

Related facts

 * Finitary alternating group is monolith in symmetric group
 * Finitary symmetric group is characteristic in symmetric group