Finitary symmetric group is conjugacy-closed in symmetric group

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.