# Difference between revisions of "Composition series"

composition series

A composition sequence 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

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