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