Finitary symmetric group on subset is conjugacy-closed

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This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Conjugacy-closed subgroup (?)) in a particular group or type of group (namely, Finitary symmetric group (?)).
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

Suppose S \subseteq T are sets. Let \operatorname{FSym}(S) and \operatorname{FSym}(T) denote the finitary symmetric groups on S and T respectively, with \operatorname{FSym}(S) viewed as a subgroup of \operatorname{FSym}(T): any finitary permutation on S corresponds to a finitary permutation on T that acts the same way on S and fixes T \setminus S pointwise.

Then, \operatorname{FSym}(S) is a conjugacy-closed subgroup of \operatorname{FSym}(T): any two finitary permutations of S that are conjugate in \operatorname{FSym}(T) are also conjugate in \operatorname{FSym}(S).