Composition series - Definition

In mathematics, a composition series of a group G is a chain of subgroups of G satisfying

<math>1 = H_0\triangleleft H_1\triangleleft \cdots \triangleleft H_n = G,<math>

where <math>\triangleleft<math> stands for normal subgroup, such that each quotient group H_i+1/H_i is a simple group. These quotient groups are called composition factors, and n is called the composition length.

The Jordan-Hölder theorem, named after Camille Jordan and Otto Hölder, states that any two composition series of a given group 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.

Every finite group has at least one composition series, but an infinite group may have none at all. For example, the infinite cyclic group has no composition series.

See also

• normal series