Finitary symmetric group equals center of symmetric group modulo finitary alternating group

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Statement

Let S be an infinite set, G = \operatorname{Sym}(S) be the Symmetric group (?) on S, K = \operatorname{FSym}(S) be the Finitary symmetric group (?), and H be the Finitary alternating group (?). Then, K/H is the Center (?) of G/H.

Proof

We need to pove that if \sigma \in G is such that [G,\sigma] \le H, then \sigma \in K. We do this by picking any \sigma \notin K, and show that [G,\sigma] is not contained in H.

If \sigma \notin K, \sigma must move infinitely many elements. There are two cases, that are collectively exhaustive:

  • \sigma contains infinitely many finite cycles.
  • \sigma contains an infinite cycle.

Case of infinitely many finite cycles

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Case of an infinite cycle

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