Composition series

composition sequence

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]).

#### References[edit]

[Co] | P.M. Cohn, "Universal algebra", Reidel (1981)

|

[Ku] | A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian)

#### Comments[edit]

For a universal algebra the notion of a composition series is more precisely defined as follows [Co]. Let $A$ be an $\Omega$-algebra and $E$ a subalgebra. A normal chain from $E$ to $A$ is then a finite chain of subalgebras of $A$, \\[ E = A_0 \subset A_1 \subset \cdots \subset A_m = A \\] together with a congruence $\mathfrak{A}_i$ on $A_i$ for $i=1,\ldots,m$ such that $A_{i-1}$ is precisely a $\mathfrak{A}_i$-class. There is a natural notion of refinement and isomorphism of normal chains: normal chains from $E$ to $A$ are isomorphic if and only if they are equally long and if there is a permutation $\sigma$ of $1,\ldots,m$ such that \\[ A_i/\mathfrak{A}_i \simeq A'_{\sigma(i)}/\mathfrak{A}'_{\sigma(i)}. \\] Then one has the Schreier refinement theorem to the effect that if $A$ is an $\Omega$-algebra with subalgebra $E$ such that on any subalgebra of $A$ all congruences commute, then any two normal chains from $E$ to $A$ have isomorphic refinements, and the Jordan–Hölder theorem that any two composition series from $E$ to $A$ on such an algebra are isomorphic.

A subgroup $H$ of a group $G$ is called subnormal if there is a chain of subgroups \\[ H = H_0 \subset H_1 \subset \cdots \subset H_m = G \\] such that $H_i$ is normal in $H_{i+1}$, $i=0,\ldots,m-1$. Consider the lattice of subnormal subgroups $L$ of $G$. Then a composition series for the partially ordered set $L$ defines in fact a composition series for $G$, 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[edit]

[Hu] | B. Huppert, "Endliche Gruppen", 1, Springer (1967)

|