# Local subgroup

From Groupprops

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]

## Contents

## Definition

A **local subgroup** of a group is defined as a subgroup that occurs as the normalizer of a **nontrivial** solvable subgroup. In symbols, a subgroup of is termed a **local subgroup** if there is a **nontrivial** solvable subgroup of such that .

Note that the nontriviality of is crucial to the definition.

## Facts

## Relation with other properties

### Stronger properties

### Weaker properties

## Metaproperties

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

A local subgroup of a group is also a local subgroup in any intermediate subgroup -- the solvable subgroup remains the same.