Conjugacy-closed implies focal subgroup equals derived subgroup

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., conjugacy-closed subgroup) must also satisfy the second subgroup property (i.e., subgroup whose focal subgroup equals its commutator subgroup)
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Suppose H is a conjugacy-closed subgroup of a group G. In other words, any two elements of H that are conjugate in G are in fact conjugate in H. Then, the focal subgroup of H equals its Commutator subgroup (?).

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