Finitary alternating group is conjugacy-closed 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 alternating group (?)) satisfying a particular subgroup property (namely, Conjugacy-closed subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

Statement

Suppose S is an infinite set, G is the symmetric group on S, and H is a subgroup of G comprising the even finitary permutations, i.e., the finitary alternating group on S. Then, H is a conjugacy-closed subgroup of G: if two elements of H are conjugate in G, they are conjugate in H.

In particular, H is a Conjugacy-closed normal subgroup (?) of G.

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