# Group of finite composition length

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

*This property makes sense for infinite groups. For finite groups, it is always true*

## Contents

## Definition

A group is said to have **finite composition length** if it satisfies the following equivalent conditions:

- It possesses a composition series of finite length, viz., a subnormal series such that all the successive quotients are simple groups.
- Every subnormal series (without repeated terms) can be refined to a composition series of finite length.
- It satisfies both the ascending chain condition on subnormal subgroups and the descending chain condition on subnormal subgroups.
- There is an upper bound on the length of any subnormal series for the group (this upper bound equals the composition length, i.e., the length of any composition series).

## Relation with other properties

### Stronger properties

- Finite group
- Composition series-unique group
- Composition factor-unique group
- Composition factor-permutable group
- Group whose chief series are composition series

### Weaker properties

- Group satisfying ascending chain condition on subnormal subgroups
- Group satisfying descending chain condition on subnormal subgroups
- Group satisfying ascending chain condition on normal subgroups
- Group satisfying descending chain condition on normal subgroups
- Group of finite chief length
- Group in which all subnormal subgroups have a common bound on subnormal depth

## Metaproperties

### Normal subgroups

This group property is normal subgroup-closed: any normal subgroup (and hence, any subnormal subgroup) of a group with the property, also has the property

View other normal subgroup-closed group properties

Any normal subgroup of a group of finite composition length again has finite composition length. In fact, if is a normal subgroup of a group , the composition length of equals the sum of the composition lengths of and .