# Finitary symmetric group on subset is conjugacy-closed

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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, Conjugacy-closed 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 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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## Statement

Suppose are sets. Let and denote the finitary symmetric groups on and respectively, with viewed as a subgroup of : any finitary permutation on corresponds to a finitary permutation on that acts the same way on and fixes pointwise.

Then, is a conjugacy-closed subgroup of : any two finitary permutations of that are conjugate in are also conjugate in .