# Conjugacy-closed implies focal subgroup equals derived subgroup

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## Contents

DIRECT: The fact or result stated in this article has a trivial/direct/straightforward proof provided we use the correct definitions of the terms involved
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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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## Statement

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 (?).