Finitary symmetric group is characteristic in symmetric group

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This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Finitary symmetric group (?)) satisfying a particular subgroup property (namely, Characteristic subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

Statement

The Finitary symmetric group (?) on an infinite set is a Characteristic subgroup (?) of the Symmetric group (?) on that set.

Facts used

  1. Finitary alternating group is monolith in symmetric group
  2. Monolith is characteristic
  3. Finitary symmetric group equals center of symmetric group modulo finitary alternating group
  4. Characteristicity is quotient-transitive

Proof

The proof follows directly by combining facts (1)-(4). PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]