# Finitary symmetric group is locally inner automorphism-balanced in symmetric group

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 $x \in T$.