# Marginal implies direct power-closed characteristic

From Groupprops

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

Then, is a direct power-closed characteristic subgroup of : for direct power of , the corresponding subgroup is a characteristic subgroup.

## Related facts

### Applications

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

## Facts used

## Proof

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