Composition series
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a 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, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module "M" is a finite increasing filtration of "M" by submodules 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 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 classes" 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 groups and Artinian modules. A related but distinct concept is a chief series: a composition series is a maximal "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 "G" is a subnormal series of finite length with strict inclusions, such that each "H" is a maximal proper normal subgroup of "H". Equivalently, a composition series is a subnormal series such that each factor group "H" / "H" is 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, formula_2 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 . 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 formula_3 has formula_4 and formula_5 as three different composition series. The sequences of composition factors obtained in the respective cases are formula_6 and formula_7 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 where all inclusions are strict and "J" is a maximal submodule of "J" 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"/"J" 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 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 . 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 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 "A" in an abelian category is a sequence of subobjects such that each quotient object "X" /"X" is simple (for  ). If "A" has a composition series, the integer "n" only depends on "A" and is called the length of "A".