Homomorph-containment is finite direct power-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., homomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., finite direct power-closed subgroup property)
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Statement

Suppose $H$ is a homomorph-containing subgroup of a group $G$. Let $n$ be a natural number. Then, in the $n^{th}$ direct power $G^{n}$ of $G$ (i.e., the external direct product of $G$ with itself $n$ times) the corresponding subgroup $H^{n}$ is a homomorph-containing subgroup.

Related facts

Similar facts

• Full invariance is finite direct power-closed
• Verbality is finite direct power-closed

Opposite facts

• Homomorph-containment is not direct power-closed
• Full invariance is not direct power-closed
• Verbality is not direct power-closed