# Group whose chief series are composition series

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

## Contents

## Definition

A **group whose chief series are composition series** is a group that satisfies the following equivalent conditions:

- The group has finite composition length, and finite chief length, and the two lengths are equal.
- The group has a chief series of finite length that is
*also*a composition series. - The group has a chief series of finite length, and
*every*chief series for the group is a composition series.

Note that it is not necessary for such a group that every composition series is a chief series. In fact, every composition series is a chief series if and only if the group is a T-group having finite composition length, i.e., it has finite composition length, and every subnormal subgroup is normal. This is because any subnormal series can be refined to a composition series for a group of finite composition length.