# Symmetric group on finite or cofinite subset is subset-conjugacy-closed

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

## Statement

Suppose $S \subseteq T$ are sets. Then, the symmetric group $\operatorname{Sym}(S)$ embeds naturally as a subgroup of $\operatorname{Sym}(T)$: any permutation of $S$ extends to a permutation of $T$ as the identity map on $T \setminus S$.

With this embedding, if either $S$ or $T \setminus S$ is finite, $\operatorname{Sym}(S)$ is a conjugacy-closed subgroup in $\operatorname{Sym}(T)$. In other words, if two subsets $A$ and $B$ of $\operatorname{Sym}(S)$ are conjugate via an element $g$ of $\operatorname{Sym}(T)$, there is an element $h$ of $\operatorname{Sym}(S)$ such that $hah^{-1} = gag^{-1}$ for all $a \in A$.