Composition series
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In mathematics, a composition series of a group G is a normal series
such that each Hi is a maximal proper normal subgroup of Hi+1. Equivalently, a composition series is a normal series such that each factor group Hi+1 / Hi is simple. The factor groups are called composition factors.
A normal 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 normal 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, the infinite cyclic group has no composition series.
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.
For example, the cyclic group C12 has {E, C2, C6, C12}, {E, C2, C4, C12}, and {E, C3, C6, C12} as different composition series. The factor groups are isomorphic to {C2, C3, C2}, {C2, C2, C3}, and {C3, C2, C2}, respectively.
Using the Jordan-Hölder theorem, it is straightforward task to prove the fundamental theorem of arithmetic.
Similarly, a composition series for a finite dimensional algebra A is a finite sequence of subalgebras
where all inclusions are proper and Jk - 1 is a maximal proper submodule of Jk. As for groups, every finite dimensional algebra has a composition series.
