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

From Groupprops

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 are sets. Then, the symmetric group embeds naturally as a subgroup of : any permutation of extends to a permutation of as the identity map on .

With this embedding, if either or is finite, is a conjugacy-closed subgroup in . In other words, if two subsets and of are conjugate via an element of , there is an element of such that for all .