Quotient-powering-invariance is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., quotient-powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive 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 quotient-powering-invariant subgroup |Get facts that use property satisfaction of quotient-powering-invariant subgroup | Get facts that use property satisfaction of quotient-powering-invariant subgroup|Get more facts about quotient-transitive subgroup property

Statement

Suppose $G$ is a group and $H,K$ are normal subgroups of $G$ with $H$ contained inside $K$. Suppose that $H$ is a quotient-powering-invariant subgroup of $G$ and $K/H$ is a quotient-powering-invariant subgroup of the quotient group $G/H$. Then, $K$ is a quotient-powering-invariant subgroup of $G$.

Related facts

• Powering-invariance is not quotient-transitive
• Powering-invariant over quotient-powering-invariant implies powering-invariant

Proof

The proof is fairly straightforward. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.