Marginal implies 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., marginal subgroup) must also satisfy the second subgroup property (i.e., direct power-closed characteristic subgroup)
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Statement

Suppose $G$ is a group and $H$ is a marginal subgroup of $G$, i.e., there is a collection of words such that $H$ is precisely the set of elements of $G$ by which left or right multiplication on any letter of the word does not affect the value of the word.

Then, $H$ is a direct power-closed characteristic subgroup of $G$: for direct power $G^{n}$ of $G$, the corresponding subgroup $H^{n}$ is a characteristic subgroup.

Related facts

Applications

• Center is direct power-closed characteristic: Follows by combining this fact with center is marginal.

Facts used

1. Marginality is direct power-closed
2. Marginal implies characteristic

Proof

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