# Local subgroup

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 local subgroup of a group is defined as a subgroup that occurs as the normalizer of a nontrivial solvable subgroup. In symbols, a subgroup $H$ of $G$ is termed a local subgroup if there is a nontrivial solvable subgroup $Q$ of $G$ such that $H = N_G(Q)$.

Note that the nontriviality of $Q$ is crucial to the definition.

## 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 condition
ABOUT 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.