Weak marginality is direct power-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., weakly marginal subgroup) satisfying a subgroup metaproperty (i.e., direct power-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about weakly marginal subgroup |Get facts that use property satisfaction of weakly marginal subgroup | Get facts that use property satisfaction of weakly marginal subgroup|Get more facts about direct power-closed subgroup property

Statement

Suppose $H$ is a weakly marginal subgroup of a group $G$, and $\alpha$ is a finite or infinite cardinal. Then, in the direct power $G^{\alpha }$, $H^{\alpha }$ is a weakly marginal subgroup. In fact, it is weakly marginal for the same collection of word-letter pairs for which $H$ is weakly marginal in $G$.

Related facts

• Marginality is direct power-closed