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Groupprops β

Symmetric groups on infinite sets are complete

Statement

Let S be an infinite set. The symmetric group on S, denoted \operatorname{Sym}(S), is a complete group: it is centerless and every automorphism of it is inner.

Facts used

Proof

Given: S is an infinite set, K = \operatorname{Sym}(S), \sigma is an automorphism of K.

To prove: \sigma is inner.

Proof: Let G = \operatorname{FSym}(S) be the subgroup of K comprising the finitary permutations.

  1. By fact (1), \sigma restricts to an automorphism, say \tau of G.
  2. By fact (2), the automorphism \tau of G arises from some inner automorphism, say \sigma', of K.
  3. Consider the ratio \sigma'\sigma^{-1}. The restriction of this automorphism to G is \tau\tau^{-1} which is the identity map. By fact (3), \sigma'\sigma^{-1} is the identity map on K, so \sigma = \sigma'. Thus, \sigma is inner.