Composition series

A composition series of a group G is a chain of subgroups of G satisfying

where stands for normal subgroup, such that the quotient group of each link in the chain is a simple group.

For a finite group G, such a composition series certainly exists; and the isomorphism classes of simple groups are unique, up to permutation. This is called the Jordan-Hölder theorem, named after Camille Jordan and Otto Hölder;.

See also normal series.