Baer-Schreier-Ulam theorem

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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.

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