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 (?)).


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.

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