# Powering-invariant normal subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and normal subgroup
View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup $H$ of a group $G$ is termed a powering-invariant normal subgroup if it is both a powering-invariant subgroup and a normal subgroup of the whole group. Here, powering-invariant means that for any prime number $p$ such that $G$ is powered over $p$, we have that $H$ is also powered over $p$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
rationally powered normal subgroup normal subgroup that is powered over all primes. |FULL LIST, MORE INFO
quotient-powering-invariant subgroup the quotient group is powered over any prime that the whole group is powered over. quotient-powering-invariant implies powering-invariant powering-invariant and normal not implies quotient-powering-invariant |FULL LIST, MORE INFO
finite normal subgroup finite and a normal subgroup Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
normal subgroup of finite index normal subgroup of finite index in the whole group. Quotient-powering-invariant subgroup|FULL LIST, MORE INFO