View source for Composition series

''composition series''

A finite subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c0243001.png" /> of a partially ordered set with least element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c0243002.png" /> and greatest element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c0243003.png" /> such that

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and all the intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c0243005.png" /> are simple (elementary) (cf. [[Elementary interval|Elementary interval]]). One can also speak of a composition series of an arbitrary interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c0243006.png" /> 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|Subgroup series]]) having no proper refinements (without repetition). A series

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is a composition series for the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c0243008.png" /> if and only if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c0243009.png" /> is a maximal normal subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430010.png" />.

All the factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430011.png" /> of a composition series are simple groups. Every normal series isomorphic to a composition series is a composition series itself. The [[Jordan–Hölder theorem|Jordan–Hölder theorem]] holds for composition series of groups. Composition series of rings, and more generally of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430012.png" />-groups, are defined in a similar way and have similar properties (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,   "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>



====Comments====
For a [[Universal algebra|universal algebra]] the notion of a composition series is more precisely defined as follows [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430013.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430014.png" />-algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430015.png" /> a subalgebra. A normal chain from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430016.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430017.png" /> is then a finite chain of subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430018.png" />,

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together with a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430023.png" /> is precisely a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430024.png" />-class. There is a natural notion of refinement and isomorphism of normal chains: normal chains from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430025.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430026.png" /> are isomorphic if and only if they are equally long and if there is a permutation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430028.png" />. Then one has the Schreier refinement theorem to the effect that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430029.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430030.png" />-algebra with subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430031.png" /> such that on any subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430032.png" /> all congruences commute, then any two normal chains from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430033.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430034.png" /> have isomorphic refinements, and the Jordan–Hölder theorem that any two composition series from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430036.png" /> on such an algebra are isomorphic.

A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430037.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430038.png" /> is called subnormal if there is a chain of subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430040.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430042.png" />. Consider the lattice of subnormal subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430044.png" />. Then a composition series for the partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430045.png" /> defines in fact a composition series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024300/c02430046.png" />, 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.)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR></table>