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

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## 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) Quotient-powering-invariant subgroup|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) Quotient-powering-invariant subgroup|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) Quotient-powering-invariant subgroup|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