Finitary symmetric group on infinite subset is conjugate-dense

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Conjugate-dense 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 conjugate-dense: in other words, every element of the group is conjugate to some element of the subgroup

Statement

Suppose $S\subseteq T$ are infinite sets. Let $\operatorname {FSym} (S)$ and $\operatorname {FSym} (T)$ denote the finitary symmetric groups on the sets $S$ and $T$ respectively, with $\operatorname {FSym} (S)$ viewed as a subgroup of $\operatorname {FSym} (T)$: any finitary permutation of $S$ is extended to a finitary permutation of $T$ by simply fixing all points in $T\setminus S$. Then, $\operatorname {FSym} (S)$ is conjugate-dense in $\operatorname {FSym} (T)$: any finitary permutation on $T$ is conjugate, in $\operatorname {FSym} (S)$, to a finitary permutation on $S$.

Related facts

• Symmetric group on proper subset is not conjugate-dense
• Symmetric group on finite or cofinite subset is conjugacy-closed
• Symmetric group on infinite coinfinite subset is not conjugacy-closed
• Finitary symmetric group on subset is conjugacy-closed