Intermediately local powering-invariant subgroup

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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 subgroup $H$ of a group $G$ is termed intermediately local powering-invariant in $G$ if, for every intermediate subgroup $K$ of $G$ (i.e., $H\leq K\leq G$), $H$ is a local 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				|FULL LIST, MORE INFO
periodic subgroup				|FULL LIST, MORE INFO
local divisibility-closed subgroup				|FULL LIST, MORE INFO
retract		(via local divisibility-closed)	(via local divisibility-closed)	|FULL LIST, MORE INFO
direct factor		(via retract)	(via retract)	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
intermediately powering-invariant subgroup	powering-invariant in every intermediate subgroup.			|FULL LIST, MORE INFO
local powering-invariant subgroup				|FULL LIST, MORE INFO
powering-invariant subgroup				|FULL LIST, MORE INFO

Facts

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

Formalisms

In terms of the intermediately operator

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