# Finitary symmetric group on subset is conjugacy-closed

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 $\operatorname{FSym}(S)$.