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| 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 and serves as a replacement of the direct sum decomposition of ''M'' into its simple constituents.
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| A composition series may not even 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.
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| 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]]''.
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| ==For groups==
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| 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, are there any unique features of the way this can be done?
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| More formally, a '''composition series''' of a [[group (mathematics)|group]] ''G'' is a [[subnormal series]] of finite length
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| :<math>1 = H_0\triangleleft H_1\triangleleft \cdots \triangleleft H_n = G,</math>
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| with strict inclusions, such that each ''H''<sub>''i''</sub> is a [[maximal element|maximal]] strict 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'''.
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| 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'''.
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| 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.
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| ===Uniqueness: Jordan–Hölder theorem===
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| 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 {{Harv|Birkhoff|1934}}.
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| ====Example====
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| 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]].
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| For example, the cyclic group ''C''<sub>12</sub> has
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| :<math> C_1\triangleleft C_2\triangleleft C_6 \triangleleft C_{12}</math>,
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| :<math> C_1\triangleleft C_2\triangleleft C_4\triangleleft C_{12}</math>,
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| :<math> C_1\triangleleft C_3\triangleleft C_6 \triangleleft C_{12}</math>
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| as different composition series.
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| The sequences of composition factors obtained in the respective cases are
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| : <math> C_2,C_3,C_2\ </math>
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| : <math> C_2,C_2,C_3\ </math> and
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| : <math> C_3,C_2,C_2.\ </math>
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| ==For modules==
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| 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
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| :<math>\{0\} = J_0 \subset \cdots \subset J_n = M</math>
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| 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.
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| 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 Artinan 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.
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| ==Generalization==
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| [[group with operators|Groups with a set of operators]] generalize group actions and ring actions on an group. A unified approach to both groups and modules can be followed as in {{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 identitical proofs.
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| 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.
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| ==For objects in an abelian category==
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| A '''composition series''' of an [[object (category theory)|object]] ''A'' in an [[abelian category]] is a sequence of subobjects
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| :<math>A=X_0\supsetneq X_1\supsetneq \dots \supsetneq X_n=0</math> | |
| such that each [[quotient object]] ''X<sub>i</sub>'' /''X<sub>i'' + 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>
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| == See also ==
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| * [[Krohn–Rhodes theory]], a semigroup analogue
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation
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| |title=Transfinite subgroup series
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| |authorlink=Garrett Birkhoff
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| |last=Birkhoff
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| |first=Garrett
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| |journal=[[Bulletin of the American Mathematical Society]]
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| |volume=40
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| |issue=12
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| |year=1934
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| |pages=847–850
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| |url=http://projecteuclid.org/euclid.bams/1183497873
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| |doi=10.1090/S0002-9904-1934-05982-2
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| }}
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| *{{Citation | last1=Isaacs | first1=I. Martin | title=Algebra: A Graduate Course | publisher=Brooks/Cole | isbn=978-0-534-19002-6 | year=1994}}
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| *{{Citation
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| | last=Kashiwara
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| | first=Masaki
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| | last2=Schapira
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| | first2=Pierre
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| | title=Categories and sheaves
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| | year=2006
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| }}
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| [[Category:Subgroup series]]
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| [[Category:Module theory]]
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