Editing Composition series

{{Short description|Decomposition of an algebraic structure}}

In [[abstract algebra]], a '''composition series''' provides a way to break up an [[algebraic structure]], such as a [[group (mathematics)|group]] or a [[module (mathematics)|module]], into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not [[semisimple module|semisimple]], hence cannot be decomposed into a [[Direct sum of modules|direct sum]] of [[simple module]]s. A composition series of a module ''M'' is a finite increasing [[filtration (abstract algebra)|filtration]] of ''M'' by [[submodule]]s such that the successive quotients are [[Simple (abstract algebra)|simple]] and serves as a replacement of the direct sum decomposition of ''M'' 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 name '''Jordan–Hölder theorem''' asserts that whenever composition series exist, the ''[[isomorphism class]]es'' 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 group]]s and [[Artinian module]]s.

A related but distinct concept is a [[chief series]]: a composition series is a maximal [[subnormal series|''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 (mathematics)|group]] ''G'' is a [[subnormal series]] of finite length
:<math>1 = H_0\triangleleft H_1\triangleleft \cdots \triangleleft H_n = G,</math>
with strict inclusions, such that each ''H''<sub>''i''</sub> is a [[maximal subgroup|maximal]] proper normal subgroup of ''H''<sub>''i''+1</sub>. Equivalently, a composition series is a subnormal series such that each factor group ''H''<sub>''i''+1</sub> / ''H''<sub>''i''</sub> is [[simple group|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, <math>\mathbb{Z}</math> 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 induction|transfinite]] ''ascending'' composition series, but not transfinite ''descending'' composition series {{Harv|Birkhoff|1934}}. 
{{harvtxt|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 <math>C_{12}</math> has <math>C_1\triangleleft C_2\triangleleft C_6 \triangleleft C_{12}, \ \, C_1\triangleleft C_2\triangleleft C_4\triangleleft C_{12}, </math> and <math>C_1\triangleleft C_3\triangleleft C_6 \triangleleft C_{12}</math> as three different composition series. The sequences of composition factors obtained in the respective cases are <math>C_2,C_3,C_2, \ \, C_2,C_2,C_3, </math> and <math>C_3,C_2,C_2.</math>

==For modules==
{{see also|Length of a module}}
The definition of composition series for modules restricts all attention to submodules, ignoring all additive subgroups that are ''not'' submodules. Given a ring ''R'' and an ''R''-module ''M'', a composition series for ''M'' is a series of submodules 

:<math>\{0\} = J_0 \subset \cdots \subset J_n = M</math>

where all inclusions are strict and ''J''<sub>''k''</sub> is a maximal submodule of ''J''<sub>''k''+1</sub> for each ''k''. As for groups,  if ''M'' has a composition series at all, then any finite strictly increasing series of submodules of ''M'' may be refined to a composition series, and any two composition series for ''M'' are equivalent. In that case, the (simple) quotient modules ''J''<sub>''k''+1</sub>/''J''<sub>''k''</sub> are known as the '''composition factors''' of ''M,'' and the Jordan–Hölder theorem holds, ensuring that the number of occurrences of each isomorphism type of simple ''R''-module as a composition factor does not depend on the choice of composition series. 

It is well known{{sfn|Isaacs|1994|loc=p.146}} that a module has a finite composition series if and only if it is both an [[Artinian module]] and a [[Noetherian module]]. If ''R'' is an [[Artinian ring]], then every finitely generated ''R''-module is Artinian and Noetherian, and thus has a finite composition series. In particular, for any field ''K'', any finite-dimensional module for a finite-dimensional algebra over ''K'' has a composition series, unique up to equivalence.

==Generalization==
[[group with operators|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 {{harv|Bourbaki|1974|loc=Ch. 1}} or {{harv|Isaacs|1994|loc=Ch. 10}}, simplifying some of the exposition. The group ''G'' 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 that ''G'' is acting on itself. An important example of this is when elements of ''G'' act by conjugation, so that the set of operators consists of the [[inner automorphism]]s. A composition series under this action is exactly a [[chief series]]. Module structures are a case of Ω-actions where Ω is a ring and some additional axioms are satisfied.

==For objects in an abelian category==
A '''composition series''' of an [[object (category theory)|object]] ''A'' in an [[abelian category]] is a sequence of subobjects
:<math>A=X_0\supsetneq X_1\supsetneq \dots \supsetneq X_n=0</math>
such that each [[quotient object]] ''X<sub>i</sub>''&nbsp;/''X''<sub>''i''&nbsp;+&nbsp;1</sub> is [[simple object|simple]] (for {{nowrap|0 ≤ ''i'' < ''n''}}). If ''A'' has a composition series, the [[integer]] ''n'' only depends on ''A'' and is called the [[length of an object|length]] of ''A''.<ref>{{harvnb|Kashiwara|Schapira|2006|loc=exercise 8.20}}</ref>

== See also ==
* [[Krohn–Rhodes theory]], a semigroup analogue
* [[Schreier refinement theorem]], any two [[subnormal series]] have equivalent composition series refinements
* [[Zassenhaus lemma]], used to prove the Schreier Refinement Theorem

==Notes==
{{reflist}}

==References==
*{{citation
|title=Transfinite subgroup series
|authorlink=Garrett Birkhoff  
|last=Birkhoff
|first=Garrett
|journal=[[Bulletin of the American Mathematical Society]]
|volume=40
|issue=12
|year=1934
|pages=847–850
|url=http://projecteuclid.org/euclid.bams/1183497873
|doi=10.1090/S0002-9904-1934-05982-2
|doi-access=free
}}
* {{citation|author=Baumslag|first=Benjamin|title=A simple way of proving the Jordan-Hölder-Schreier     theorem|journal=American Mathematical Monthly|volume=113|year=2006|issue=10|
pages=933–935|doi=10.2307/27642092}}
*{{Citation | last1=Bourbaki | first1=N. | title=Algebra | publisher=Hermann, Paris; Addison-Wesley Publishing Co., Reading Mass. | year=1974}}
*{{Citation | last1=Isaacs | first1=I. Martin | title=Algebra: A Graduate Course | publisher=Brooks/Cole | isbn=978-0-534-19002-6 | year=1994}}
*{{Citation
| last=Kashiwara
| first=Masaki
| last2=Schapira
| first2=Pierre
| title=Categories and sheaves
| year=2006
}}

[[Category:Subgroup series]]
[[Category:Module theory]]