# Intermediately local powering-invariant subgroup

From Groupprops

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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 of a group is termed **intermediately local powering-invariant** in if, for every intermediate subgroup of (i.e., ), is a local powering-invariant subgroup of .

## 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 | Intermediately powering-invariant subgroup|FULL LIST, MORE INFO |

## Facts

## 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