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			|FULL LIST, MORE INFO
normal subgroup of finite index	normal subgroup of finite index in the whole group.			|FULL LIST, MORE INFO
normal subgroup contained in the hypercenter				|FULL LIST, MORE INFO
endomorphism kernel	normal subgroup that is the kernel of an endomorphism	(via quotient-powering-invariant)	any normal subgroup of a finite group that is not an endomorphism kernel works.	|FULL LIST, MORE INFO
complemented normal subgroup	normal subgroup with a (possibly non-normal) complement	[(via endomorphism kernel)		|FULL LIST, MORE INFO
direct factor	normal subgroup with normal complement	(via complemented normal)	(via complemented normal)	|FULL LIST, MORE INFO
characteristic subgroup of abelian group	characteristic subgroup and the whole group is an abelian group.	Characteristic subgroup of abelian group is powering-invariant and characteristic implies normal		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
powering-invariant subgroup
normal subgroup