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)

Comments

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

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