# Composition series

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composition series

A finite subset of a partially ordered set with least element and greatest element such that

and all the intervals are simple (elementary) (cf. Elementary interval). One can also speak of a composition series of an arbitrary interval 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

is a composition series for the group if and only if every is a maximal normal subgroup in .

All the factors 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 -groups, are defined in a similar way and have similar properties (see [2]).

#### References

 [1] P.M. Cohn, "Universal algebra" , Reidel (1981) [2] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)