Subgroup series | Module theory

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. (Wikipedia).

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Related pages

Module (mathematics) | Algebraic structure | If and only if | Chief series | Direct sum of modules | Up to | Isomorphism | Group (mathematics) | Isomorphism class | Noetherian module | Permutation | Artinian ring | Semisimple module | Simple (abstract algebra) | Schreier refinement theorem | Transfinite induction | Maximal subgroup | Krohn–Rhodes theory | Finite group | Simple group | Quotient object | Abelian category | Infinite group | Integer | Fundamental theorem of arithmetic | Simple module | Artinian module | Cyclic group | Normal subgroup | Zassenhaus lemma | Abstract algebra | Group with operators | Camille Jordan | Inner automorphism