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.
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.
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.
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 G is a subnormal series of finite length with strict inclusions, such that each Hi is a maximal proper normal subgroup of Hi+1.
That is, there are no additional subgroups which can be "inserted" into a 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.
The Jordan–Hölder theorem is also true for transfinite ascending composition series, but not transfinite descending composition series (Birkhoff 1934).
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.
The sequences of composition factors obtained in the respective cases are
In that case, the (simple) quotient modules Jk+1/Jk 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.
A unified approach to both groups and modules can be followed as in (Bourbaki 1974, Ch.
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.
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.
Module structures are a case of Ω-actions where Ω is a ring and some additional axioms are satisfied.
If A has a composition series, the integer n only depends on A and is called the length of A.