Finitary symmetric group is locally inner automorphism-balanced 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 symmetric group (?)) satisfying a particular subgroup property (namely, Locally inner automorphism-balanced subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

Statement

Suppose S is a set, G is the symmetric group \operatorname{Sym}(S) on S, and H is the finitary symmetric group \operatorname{FSym}(S) on S, viewed as a subgroup of G. Then, H is a locally inner automorphism-balanced subgroup of G. In other words, for any g \in G, the restriction of the inner automorphism x \mapsto gxg^{-1} of G to H is a locally inner automorphism of H, i.e., for any finite subset T of H, there exists h \in H such that hxh^{-1} = gxg^{-1} for all x \in T.