Normal subgroup satisfying the subgroup-to-quotient powering-invariance implication

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Definition

Suppose $H$ is a normal subgroup of a group $G$. We say that $H$ satisfies a subgroup-to-quotient powering-invariance implication if for any prime $p$ such that both $G$ and $H$ are powered over $p$, the quotient group $G/H$ is also powered over $p$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
quotient-powering-invariant subgroup		(by definition)		|FULL LIST, MORE INFO
finite normal subgroup	normal subgroup that is finite as a group	via quotient-powering-invariant	(via quotient-powering-invariant)	|FULL LIST, MORE INFO
normal subgroup of finite index	normal subgroup such that the quotient group is finite	via quotient-powering-invariant	(via quotient-powering-invariant)	|FULL LIST, MORE INFO
complemented normal subgroup	normal subgroup with a permutable complement, i.e., normal part of an internal semidirect product	via quotient-powering-invariant	(via quotient-powering-invariant)	|FULL LIST, MORE INFO
nilpotent-quotient subgroup	normal subgroup such that the quotient group is a nilpotent group	nilpotent-quotient implies subgroup-to-quotient powering-invariance implication		|FULL LIST, MORE INFO
normal subgroup contained in the hypercenter	contained in the hypercenter	normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication		|FULL LIST, MORE INFO