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 ).
|||P.M. Cohn, "Universal algebra" , Reidel (1981)|
|||A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)|
For a universal algebra the notion of a composition series is more precisely defined as follows . Let be an -algebra and a subalgebra. A normal chain from to is then a finite chain of subalgebras of ,
together with a congruence on for such that is precisely a -class. There is a natural notion of refinement and isomorphism of normal chains: normal chains from to are isomorphic if and only if they are equally long and if there is a permutation of such that . Then one has the Schreier refinement theorem to the effect that if is an -algebra with subalgebra such that on any subalgebra of all congruences commute, then any two normal chains from to have isomorphic refinements, and the Jordan–Hölder theorem that any two composition series from to on such an algebra are isomorphic.
A subgroup of a group is called subnormal if there is a chain of subgroups such that is normal in , . Consider the lattice of subnormal subgroups of . Then a composition series for the partially ordered set defines in fact a composition series for , and vice versa. Something analogous can be formulated for universal algebras (These statements of course do not hold for, respectively, the lattice of normal subgroups and the lattice of congruences.)
|[a1]||B. Huppert, "Endliche Gruppen" , 1 , Springer (1967)|
Composition series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Composition_series&oldid=13738