# Chief series

In abstract algebra, a **chief series** is a maximal normal series for a group.

It is similar to a composition series, though the two concepts are distinct in general: a chief series is a maximal *normal* series, while a composition series is a maximal *subnormal* series.

Chief series can be thought of as breaking the group down into less complicated pieces, which may be used to characterize various qualities of the group.

## Definition

A chief series is a maximal normal series for a group. Equivalently, a chief series is a composition series of the group *G* under the action of inner automorphisms.

In detail, if *G* is a group, then a **chief series** of *G* is a finite collection of normal subgroups *N*_{i} ⊆ *G*,

such that each quotient group *N*_{i+1}/*N*_{i}, for *i* = 1, 2,..., *n* − 1, is a minimal normal subgroup of *G*/*N*_{i}. Equivalently, there does not exist any subgroup *A* normal in *G* such that *N*_{i} < *A* < *N*_{i+1} for any *i*. In other words, a chief series may be thought of as "full" in the sense that no normal subgroup of *G* may be added to it.

The factor groups *N*_{i+1}/*N*_{i} in a chief series are called the **chief factors** of the series. Unlike composition factors, chief factors are not necessarily simple. That is, there may exist a subgroup *A* normal in *N*_{i+1} with *N*_{i} < *A* < *N*_{i+1}, but *A* is not normal in *G*. However, the chief factors are always characteristically simple, that is, they have no proper nontrivial characteristic subgroups. In particular, a finite chief factor is a direct product of isomorphic simple groups.

## Properties

### Existence

Finite groups always have a chief series, though infinite groups need not have a chief series. For example, the group of integers **Z** with addition as the operation does not have a chief series. To see this, note **Z** is cyclic and abelian, and so all of its subgroups are normal and cyclic as well. Supposing there exists a chief series *N*_{i} leads to an immediate contradiction: *N*_{1} is cyclic and thus is generated by some integer *a*, however the subgroup generated by 2*a* is a nontrivial normal subgroup properly contained in *N*_{1}, contradicting the definition of a chief series.

### Uniqueness

When a chief series for a group exists, it is generally not unique. However, a form of the Jordan–Hölder theorem states that the chief factors of a group are unique up to isomorphism, independent of the particular chief series they are constructed from. In particular, the number of chief factors is an invariant of the group *G*, as well as the isomorphism classes of the chief factors and their multiplicities.

### Other properties

In abelian groups, chief series and composition series are identical, as all subgroups are normal.

Given any normal subgroup *N* ⊆ *G*, one can always find a chief series in which *N* is one of the elements (assuming a chief series for *G* exists in the first place.) Also, if *G* has a chief series and *N* is normal in *G*, then both *N* and *G*/*N* have chief series. The converse also holds: if *N* is normal in *G* and both *N* and *G*/*N* have chief series, *G* has a chief series as well.

## References

- Isaacs, I. Martin (1994).
*Algebra: A Graduate Course*. Brooks/Cole. ISBN 0-534-19002-2.