# Composition series

A composition series is a finite subset $\{a_0,\ldots,a_n\}$ of a partially ordered set with least element $0$ and greatest element $1$ such that $0 = a_0 < a_1 < \cdots < a_n = 1$ and all the intervals $[a_i,a_{i+1}]$ are simple (elementary) (cf. Elementary interval). One can also speak of a composition series of an arbitrary interval $[a,b]$ of a partially ordered set. Composition series certainly do not always exists.
A composition series of a universal algebra is defined in terms of congruences. Since congruences in groups are defined by normal subgroups, a composition series of a group can be defined as a normal series of it (see Subgroup series) having no proper refinements (without repetition). A series $E = G_0 \subset \cdots \subset G_{k-1} \subset G_k = G$ is a composition series for the group $G$ if and only if every $G_{i-1}$ is a maximal normal subgroup in $G_i$.
All the factors $G_i/G_{i-1}$ of a composition series are simple groups. Every normal series isomorphic to a composition series is a composition series itself. The Jordan–Hölder theorem holds for composition series of groups. Composition series of rings, and more generally of $\Omega$-groups, are defined in a similar way and have similar properties (see [Ku]).