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