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Fully invariant implies finite direct power-closed characteristic

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., fully invariant subgroup) must also satisfy the second subgroup property (i.e., finite direct power-closed characteristic subgroup)
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Contents

Statement

Related facts

Facts used

Proof

The proof follows directly from facts (1) and (2).