Inabstract algebra, acomposition series provides a way to break up analgebraic structure, such as agroup or amodule, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are notsemisimple, hence cannot be decomposed into adirect sum ofsimple modules. A composition series of a moduleM is a finite increasingfiltration ofM bysubmodules such that the successive quotients aresimple and serves as a replacement of the direct sum decomposition ofM into its simple constituents.

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

A related but distinct concept is achief series: a composition series is a maximalsubnormal series, while a chief series is a maximalnormal series.

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

More formally, acomposition series of agroupG is asubnormal series of finite length

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

with strict inclusions, such that eachH_i is amaximal proper normal subgroup ofH_i+1. Equivalently, a composition series is a subnormal series such that each factor groupH_i+1 /H_i issimple. The factor groups are calledcomposition factors.

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

If a composition series exists for a groupG, then any subnormal series ofG can berefined to a composition series, informally, by inserting subgroups into the series up to maximality. Everyfinite group has a composition series, but not everyinfinite group has one. For example,${\displaystyle \mathbb{Z}}$ has no composition series.

A group may have more than one composition series. However, theJordan–Hölder theorem (named afterCamille Jordan andOtto 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 topermutation andisomorphism. This theorem can be proved using theSchreier refinement theorem. The Jordan–Hölder theorem is also true fortransfiniteascending composition series, but not transfinitedescending 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.

For acyclic group of ordern, composition series correspond to ordered prime factorizations ofn, and in fact yields a proof of thefundamental 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${\displaystyle C_3,C_2,C_2.}$

The definition of composition series for modules restricts all attention to submodules, ignoring all additive subgroups that arenot submodules. Given a ringR and anR-moduleM, a composition series forM is a series of submodules

${\displaystyle \{0\}=J_0\subset \cdots \subset J_n=M}$

where all inclusions are strict andJ_k is a maximal submodule ofJ_k+1 for eachk. As for groups, ifM has a composition series at all, then any finite strictly increasing series of submodules ofM may be refined to a composition series, and any two composition series forM are equivalent. In that case, the (simple) quotient modulesJ_k+1/J_k are known as thecomposition factors ofM, and the Jordan–Hölder theorem holds, ensuring that the number of occurrences of each isomorphism type of simpleR-module as a composition factor does not depend on the choice of composition series.

It is well known^[1] that a module has a finite composition series if and only if it is both anArtinian module and aNoetherian module. IfR is anArtinian ring, then every finitely generatedR-module is Artinian and Noetherian, and thus has a finite composition series. In particular, for any fieldK, any finite-dimensional module for a finite-dimensional algebra overK has a composition series, unique up to equivalence.

Groups with a set of operators generalize group actions and ring actions on a group. A unified approach to both groups and modules can be followed as in (Bourbaki 1974, Ch. 1) or (Isaacs 1994, Ch. 10), simplifying some of the exposition. The groupG is viewed as being acted upon by elements (operators) from a set Ω. Attention is restricted entirely to subgroups invariant under the action of elements from Ω, called Ω-subgroups. Thus Ω-composition series must use only Ω-subgroups, and Ω-composition factors need only be Ω-simple. The standard results above, such as the Jordan–Hölder theorem, are established with nearly identical proofs.

The special cases recovered include when Ω =G so thatG is acting on itself. An important example of this is when elements ofG act by conjugation, so that the set of operators consists of theinner automorphisms. A composition series under this action is exactly achief series. Module structures are a case of Ω-actions where Ω is a ring and some additional axioms are satisfied.

Acomposition series of anobjectA in anabelian category is a sequence of subobjects

${\displaystyle A=X_0⊋X_1⊋\cdots ⊋X_n=0}$

such that eachquotient objectX_i /X_i + 1 issimple (for0 ≤i <n). IfA has a composition series, theintegern only depends onA and is called thelength ofA.^[2]

• Krohn–Rhodes theory, a semigroup analogue
• Schreier refinement theorem, any twosubnormal series have equivalent composition series refinements
• Zassenhaus lemma, used to prove the Schreier Refinement Theorem

• Birkhoff, Garrett (1934),"Transfinite subgroup series",Bulletin of the American Mathematical Society,40 (12):847–850,doi:10.1090/S0002-9904-1934-05982-2
• Baumslag, Benjamin (2006), "A simple way of proving the Jordan-Hölder-Schreier theorem",American Mathematical Monthly,113 (10):933–935,doi:10.2307/27642092
• Bourbaki, N. (1974),Algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading Mass.
• Isaacs, I. Martin (1994),Algebra: A Graduate Course, Brooks/Cole,ISBN 978-0-534-19002-6
• Kashiwara, Masaki; Schapira, Pierre (2006),Categories and sheaves

• Subgroup series
• Module theory

• Articles with short description
• Short description matches Wikidata