Composition series

A composition series may not exist, and when it does, it need not be unique. Nevertheless, a group of results known under the general name Jordan–Hölder theorem asserts that whenever composition series exist, the isomorphism classes of simple pieces (although, perhaps, not their location in the composition series in question) and their multiplicities are uniquely determined. Composition series may thus be used to define invariants of finite groups and Artinian modules.

A related but distinct concept is a chief series: a composition series is a maximal subnormal series, while a chief series is a maximal normal series.

For groups

If a group G has a normal subgroup N, then the factor group G/N may be formed, and some aspects of the study of the structure of G may be broken down by studying the "smaller" groups G/N and N. If G has no normal subgroup that is different from G and from the trivial group, then G is a simple group. Otherwise, the question naturally arises as to whether G can be reduced to simple "pieces", and if so, whether there are any unique features of the way this can be done.

More formally, a composition series of a group G is a subnormal series of finite length

${\displaystyle 1=H_0◃H_1◃\cdots ◃H_n=G,}$

with strict inclusions, such that each H_i is a maximal proper normal subgroup of H_i+1. Equivalently, a composition series is a subnormal series such that each factor group H_i+1 / H_i is simple. The factor groups are called composition factors.

A subnormal series is a composition series if and only if it is of maximal length. That is, there are no additional subgroups which can be "inserted" into a composition series. The length n of the series is called the composition length.

If a composition series exists for a group G, then any subnormal series of G can be refined to a composition series, informally, by inserting subgroups into the series up to maximality. Every finite group has a composition series, but not every infinite group has one. For example, ${\displaystyle \mathbb{Z}}$ has no composition series.

Uniqueness: Jordan–Hölder theorem

A group may have more than one composition series. However, the Jordan–Hölder theorem (named after Camille Jordan and Otto Hölder) states that any two composition series of a given group are equivalent. That is, they have the same composition length and the same composition factors, up to permutation and isomorphism. This theorem can be proved using the Schreier refinement theorem. The Jordan–Hölder theorem is also true for transfinite ascending composition series, but not transfinite descending composition series (Birkhoff 1934). Baumslag (2006) gives a short proof of the Jordan–Hölder theorem by intersecting the terms in one subnormal series with those in the other series.

Example

For a cyclic group of order n, composition series correspond to ordered prime factorizations of n, and in fact yields a proof of the fundamental theorem of arithmetic.

For example, the cyclic group ${\displaystyle C_{12}}$ has ${\displaystyle C_1◃C_2◃C_6◃C_{12},C_1◃C_2◃C_4◃C_{12},}$ and ${\displaystyle C_1◃C_3◃C_6◃C_{12}}$ as three different composition series. The sequences of composition factors obtained in the respective cases are ${\displaystyle C_2,C_3,C_2,C_2,C_2,C_3,}$ and ${\displaystyle C_3,C_2,C_2.}$