Finitary symmetric group is conjugacy-closed in symmetric group

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This article gives the statement, and proof, of a particular subgroup in a group being conjugacy-closed: in other words, any two elements of the subgroup that are conjugate in the whole group, are also conjugate in the subgroup
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Statement

The finitary symmetric group on a set is a conjugacy-closed subgroup (in fact, a conjugacy-closed normal subgroup) inside the symmetric group on that set. In other words, any inner automorphism in the symmetric group on a set, restricts to a class-preserving automorphism in the finitary symmetric group.

More explicitly, if \sigma is an arbitrary permutation on a set S, and \alpha is a finitary permutation on S, then there exists a finitary permutation \beta on S, such that \sigma \alpha\sigma^{-1} = \beta\alpha\beta^{-1}.

Related facts

  • Class-preserving not implies inner: Not every class-preserving automorphism of a group is inner. In fact, any element of the symmetric group that is not in the finitary symmetric group gives a class-preserving automorphism of the finitary symmetric group, that is not inner. (The fact that it isn't inner follows from the fact that the finitary symmetric group is centralizer-free inside the symmetric group).

Proof

The proof involves two steps:

  1. Conjugation by any element in the symmetric group preserves the cycle type of a finitary permutation.
  2. Given two finitary permutations with the same cycle type, there exists a finitary permutation taking the first to the second.