# Intermediately 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]

## 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., $H \le K \le G$), $H$ 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 Intermediately local powering-invariant subgroup|FULL LIST, MORE INFO
periodic subgroup Intermediately local powering-invariant 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 Divisibility-closed subgroup, Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup, Local powering-invariant subgroup|FULL LIST, MORE INFO
retract has a normal complement (via local divisibility-closed) (via local divisibility-closed) Divisibility-closed subgroup, Endomorphism image, Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup, Local divisibility-closed subgroup, Local powering-invariant subgroup, Verbally closed subgroup|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) Intermediately local powering-invariant subgroup|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

## 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