Marginal implies direct power-closed characteristic

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.

Applications

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

Facts used

 * 1) uses::Marginality is direct power-closed
 * 2) uses::Marginal implies characteristic

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