Finitary alternating group is conjugacy-closed in symmetric group

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 $G$.

Related facts

• Finitary symmetric group is conjugacy-closed in symmetric group
• Finitary alternating group is monolith in symmetric group
• Finitary alternating group is characteristic in symmetric group