Quotient-powering-invariant subgroup

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]

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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 $p$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup of finite group
normal subgroup of periodic group
normal subgroup of finite index		normal of finite index implies quotient-powering-invariant
finite normal subgroup		finite normal implies quotient-powering-invariant

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
powering-invariant subgroup		quotient-powering-invariant implies powering-invariant	powering-invariant not implies quotient-powering-invariant	|FULL LIST, MORE INFO