en>Magioladitis: WPCleaner v1.34 - Fixed using WP:WCW (Reference before punctuation)

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WPCleaner v1.34 - Fixed using WP:WCW (Reference before punctuation)

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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~~.

	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.		Feel free to visit my web site - [http://prometeu.net clash of clans hack tool]

	~~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, are there 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 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'''.

	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]] ''ascending'' composition series, but not transfinite ''descending'' composition series {{Harv\|Birkhoff\|1934}}.

	~~====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 ''C''<sub>12</sub> has~~
	~~:<math> C_1\triangleleft C_2\triangleleft C_6 \triangleleft C_{12}</math>,~~
	~~:<math> C_1\triangleleft C_2\triangleleft C_4\triangleleft C_{12}</math>,~~
	~~:<math> C_1\triangleleft C_3\triangleleft C_6 \triangleleft C_{12}</math>~~
	~~as different composition series.~~
	~~The sequences of composition factors obtained in the respective cases are~~
	~~: <math> C_2,C_3,C_2\ </math>~~
	~~: <math> C_2,C_2,C_3\ </math> and~~
	~~: <math> C_3,C_2,C_2.\ </math>~~

	~~==For modules==~~
	~~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 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.

	~~==Generalization==~~
	[[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.

	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>'' ~~/~~''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>~~

	~~== See also ==~~
	* [[Krohn–Rhodes theory]], a semigroup analogue

	~~==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~~
	}}
	*{{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]~~]

en>Citation bot: [442]Add: pmid, issue, doi. fconaway-activated.

2013-12-15T21:59:27Z

[442]Add: pmid, issue, doi. fconaway-activated.

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122.60.48.79: /* Introduction */

2012-08-21T12:15:37Z

Introduction

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