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In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a composition series provides a way to break up an
algebraic structure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, such as a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
or a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
, 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, hence cannot be decomposed into a
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of
simple module In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
s. A composition series of a module ''M'' is a finite increasing
filtration Filtration is a physical separation process A separation process is a method that converts a mixture or solution of chemical substances into two or more distinct product mixtures. At least one of results of the separation is enriched in one or ...
of ''M'' by
submodule In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s such that the successive quotients are
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * Simple (album), ''Simple'' (album), by Andy Yorke, 2008, and its title track * Simple (Florida Georgia Line song), "Simple" (Florida Ge ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
s and
Artinian module In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
s. A related but distinct concept is a
chief seriesIn abstract algebra, a chief series is a maximal normal series for a group (mathematics), group. It is similar to a composition series, though the two concepts are distinct in general: a chief series is a maximal ''normal'' series, while a compositi ...
: 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 A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
''G'' is a subnormal series of finite length :1 = H_0\triangleleft H_1\triangleleft \cdots \triangleleft H_n = G, with strict inclusions, such that each ''H''''i'' is a maximal subgroup, 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 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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
has a composition series, but not every infinite group has one. For example, \mathbb 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 . 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 C_ has C_1\triangleleft C_2\triangleleft C_6 \triangleleft C_, \ \, C_1\triangleleft C_2\triangleleft C_4\triangleleft C_, and C_1\triangleleft C_3\triangleleft C_6 \triangleleft C_ as three different composition series. The sequences of composition factors obtained in the respective cases are C_2,C_3,C_2, \ \, C_2,C_2,C_3, and C_3,C_2,C_2.


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 :\ = J_0 \subset \cdots \subset J_n = M where all inclusions are strict and ''J''''k'' is a maximal submodule of ''J''''k''+1 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''''k''+1/''J''''k'' 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 that a module has a finite composition series if and only if it is both an
Artinian module In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
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 or , 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 automorphisms. A composition series under this action is exactly a
chief seriesIn abstract algebra, a chief series is a maximal normal series for a group (mathematics), group. It is similar to a composition series, though the two concepts are distinct in general: a chief series is a maximal ''normal'' series, while a compositi ...
. 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 :A=X_0\supsetneq X_1\supsetneq \dots \supsetneq X_n=0 such that each quotient object ''Xi'' /''X''''i'' + 1 is simple object, simple (for ). If ''A'' has a composition series, the integer ''n'' only depends on ''A'' and is called the length of an object, length of ''A''.


See also

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


Notes


References

* * * * *{{Citation , last=Kashiwara , first=Masaki , last2=Schapira , first2=Pierre , title=Categories and sheaves , year=2006 Subgroup series Module theory