Finitary alternating groups are simple

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Definition

Let S be an infinite set. The Finitary alternating group (?) on S, i.e., the group of even finitary permutations on S under composition, is a simple group.

Facts used

  1. Alternating groups are simple: The alternating group on a finite set is simple when the set has at least 5 elements.
  2. Simplicity is directed union-closed: A directed union of simple subgroups is simple.

Proof

The proof follows from facts (1) and (2), and the observation that the finitary alternating group on an infinite set is the directed union of alternating groups on all finite subsets of size at least 5.