# Composition 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

### Symbol-free definition

A subgroup of a group is termed a **composition subgroup** if there is a composition series starting at the subgroup and ending at the whole group.

## Relation with other properties

### Stronger properties

- Sub-(maximal normal) subgroup:
`For full proof, refer: Sub-(maximal normal) implies composition`

### Weaker properties

- Serial subgroup:
`For full proof, refer: Composition implies serial`

## 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

Any composition subgroup of a composition subgroup is a composition subgroup -- we can concatenate the two composition series.