Quotient-powering-invariant subgroup

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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 satisfying the subgroup-to-quotient powering-invariance implication
View other subgroup property conjunctions | view all subgroup properties

Definition

A normal subgroup $H$ of a group $G$ is termed a quotient-powering-invariant subgroup if, for any prime number $p$ such that $G$ is a powered for $p$, the quotient group $G/H$ is also powered for $p$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
quotient-transitive subgroup property	Yes	quotient-powering-invariance is quotient-transitive	If $H\leq K\leq G$ are such that $H$ is quotient-powering-invariant in $G$ and $K/H$ is quotient-powering-invariant in $G/H$, then $K$ is quotient-powering-invariant in $G$.
union-closed subgroup property	Yes	quotient-powering-invariance is union-closed	If $H_{i},i\in I$ are all quotient-powering-invariant subgroups of a group $G$, and their set-theoretic union $\bigcup _{i\in I}H_{i}$ is a subgroup $H$, then $H$ is also a quotient-powering-invariant subgroup of $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup of finite group	the whole group is finite
normal subgroup of periodic group	every element in the whole group has finite order
normal subgroup of finite index	the quotient group is finite	normal of finite index implies quotient-powering-invariant
finite normal subgroup	the normal subgroup is finite	finite normal implies quotient-powering-invariant
endomorphism kernel	normal subgroup that is the kernel of an endomorphism	endomorphism kernel implies 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, see also direct proof)		|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 quotient-powering-invariant		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
powering-invariant normal subgroup		quotient-powering-invariant implies powering-invariant	powering-invariant and normal not implies quotient-powering-invariant	|FULL LIST, MORE INFO
powering-invariant subgroup		quotient-powering-invariant implies powering-invariant	(via powering-invariant normal)	|FULL LIST, MORE INFO
normal subgroup satisfying the subgroup-to-quotient powering-invariance implication				|FULL LIST, MORE INFO
normal subgroup		(by definition)	(via powering-invariant normal)	|FULL LIST, MORE INFO

Properties whose conjunction with powering-invariance implies quotient-powering-invariance

The relevant subgroup property is normal subgroup satisfying the subgroup-to-quotient powering-invariance implication. In fact, the conjunction of this with powering-invariant subgroup precisely gives quotient-powering-invariant subgroup.

This property is implied both by being a central subgroup and by being a normal subgroup contained in the hypercenter.