Composition series
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In mathematics, a composition series of a group G is a normal series
- <math>1 = H_0\triangleleft H_1\triangleleft \cdots \triangleleft H_n = G,<math>
such that each Hi is a maximal 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.
In general, a group will have multiple, different 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.pl:CiÄ…g_kompozycyjny_grupy