Intermediately powering-invariant subgroup

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Definition

A subgroup $H$ of a group $G$ is termed an intermediately powering-invariant subgroup if, for any subgroup $K$ of $G$ containing $H$ (i.e., $H\leq K\leq G$), $H$ is a powering-invariant subgroup of $K$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite subgroup		follows from finite implies powering-invariant		|FULL LIST, MORE INFO
periodic subgroup				|FULL LIST, MORE INFO
subgroup of finite index		follows from finite index implies powering-invariant		|FULL LIST, MORE INFO
normal subgroup of finite index		(via subgroup of finite index)	(via subgroup of finite index)	|FULL LIST, MORE INFO
characteristic subgroup of abelian group		characteristic subgroup of abelian group implies intermediately powering-invariant		|FULL LIST, MORE INFO
local divisibility-closed subgroup				|FULL LIST, MORE INFO
retract	has a normal complement	(via local divisibility-closed)	(via local divisibility-closed)	|FULL LIST, MORE INFO
complemented normal subgroup	normal subgroup with a permutable complement, i.e., part of an internal semidirect product	(via endomorphism kernel)	(via endomorphism kernel)	|FULL LIST, MORE INFO
direct factor	factor in an internal direct product	(via complemented normal, also via retract)	(via complemented normal, also via retract)	|FULL LIST, MORE INFO
intermediately local powering-invariant subgroup	local powering-invariant subgroup in every intermediate subgroup			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
powering-invariant subgroup				|FULL LIST, MORE INFO

Facts

• Local powering-invariant subgroup containing the center is intermediately powering-invariant in nilpotent group

Formalisms

In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: powering-invariant subgroup
View other properties obtained by applying the intermediately operator