Conjugacy-closedness is transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., conjugacy-closed subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement with symbols
Suppose are groups such that is conjugacy-closed in and is conjugacy-closed in .
Given: Groups such that is conjugacy-closed in and is conjugacy-closed in .
To prove: is conjugacy-closed in .
Proof: We need to show that if are conjugate in , then they are conjugate in . First, observe that since , are elements of conjugate in . Since is conjugacy-closed in , are conjugate in .
Thus, are elements of that are conjugate in . Since is conjugacy-closed in , the elements are conjugate in .