# Finitary alternating group is conjugacy-closed in symmetric group

From Groupprops

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

## Statement

Suppose is an infinite set, is the symmetric group on , and is a subgroup of comprising the even finitary permutations, i.e., the finitary alternating group on . Then, is a conjugacy-closed subgroup of : if two elements of are conjugate in , they are conjugate in .

In particular, is a Conjugacy-closed normal subgroup (?) of .