# Strongly embedded 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]

*This article is about a term related to the Classification of finite simple groups*

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed **strongly embedded** or **tightly embedded** if it has even order, is self-normalizing and its intersection with any other conjugate has odd order.

### Definition with symbols

A subgroup of a group is termed **strongly embedded** or **tightly embedded** in if has even order and for any in which is not in , ∩ has odd order.

## Relation with other properties

### Stronger properties

### Weaker properties

### Opposites

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

The condition of having even order is clearly transitive, while the condition of the intersection with any conjugate having odd order is also transitive. `For full proof, refer: Strongly embedded satisfies transitivity`