# Baer-Schreier-Ulam theorem

From Groupprops

## Statement

Let be an infinite set and denote the symmetric group on . For every cardinal , define as the group of all permutations on that move at most elements. Then, the normal subgroups of 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 for all ordinals , where is the group of permutations whose support has size equal to the cardinality of any ordinal smaller than .