Finitary symmetric group is automorphism-faithful 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, Automorphism-faithful subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

Statement

Let S be any set. Then, the finitary symmetric group \operatorname{FSym}(S) is an automorphism-faithful subgroup in \operatorname{Sym}(S): any nontrivial automorphism of \operatorname{Sym}(S) that restricts to an automorphism on \operatorname{FSym}(S) restricts to a nontrivial automorphism on \operatorname{FSym}(S).

Related facts

Facts used

  1. Finitary symmetric group is centralizer-free in symmetric group
  2. Finitary symmetric group is normal in symmetric group
  3. Normal and centralizer-free implies automorphism-faithful

Proof

The proof follows directly by piecing together facts (1)-(3).