Applications of the Jordan-Hölder Theorem.

Solution 1:

(ADDITION) Late remark: you say in your question that the only application you know is Rotman's proof of the fundamental theorem of arithmetic via Jordan-Hölder, indeed that is Corollary 4.56 of his book "Advanced Modern Algebra". Just for completeness, I believe it is worth mentioning including here his discussion on the next two pages about the significance of the Jordan-Hölder theorem. The extension problem for pairs of groups $K, Q$ is the classification of all possible groups $G$ satisfying the short exact sequence $$1\rightarrow K \hookrightarrow G \twoheadrightarrow Q \rightarrow 1\;. $$ Since the direct product $K\times Q$ is such an extension, it is customary to interpret such $G$ as "generalized products". By Rotman's Prop 4.53, every finite group $G$ has a composition series, i.e. a finite sequence of normal subgroups $$G=K_0\unrhd K_1\unrhd ...\unrhd K_n=\{1\},$$ such that the nontrivial factor groups $Q_1:=K_0/K_1,\, Q_2:=K_1/K_2,\, ...,\, Q_n:=K_{n-1}/K_n$ are simple. Jordan-Hölder guarantees that if any such composition series exist for $G$, they have the same set of nontrivial factor groups, thus being invariants of $G$ along with the length of such series. Now, Rotman discusses in pages 256-257 all this as the fundamental significance of the theorem: since $K_{i-1}$ is an extension of $K_i$ by $Q_i$ for all $i$, if we could solve the extension problem we could then recapture $K_i$ in succession up to $G=K_0$ from its factor groups $Q_n,Q_{n-1},...,Q_1$, making $G$ a generalized product of its factor groups which are uniquely determined by it; therefore the Jordan-Hölder theorem is a unique factorization theorem. As the classification theorem of finite simple groups has been proved (thousands of pages!), we could survey all finite groups via "generalized products of simple groups" by using Jordan-Hölder if we could solve the extension problem for them. (This is still unsolved and leads to group cohomology among other things).

The Jordan-Hölder theorem for groups guarantees that any composition series of a module over a ring are equivalent, so that the lengths of its longest such chains are the same. This makes length a well-defined invariant which is finite iff the module is Artinian and Noetherian. This coincides with, and generalizes, the dimension of vector spaces for general modules.

This consequence of Jordan-Hölder is a fundamental concept with applications in algebraic geometry, especially in intersection theory, where the multiplicity of intersecting components of subschemes is counted via lengths of quotient modules of their ring of regular algebraic functions. In particular if $M$ is a finitely generated module over a Noetherian ring $R$, then there is a finite chain of submodules: $$0=M_0\subsetneq M_1\subsetneq \cdots\subsetneq M_r=M\;\;\text{such that}\;\; M_i/M_{i-1}\cong R/\mathfrak p_i\;\;\text{for prime ideals}\;\; \mathfrak p_i\subset R.$$ The composition series is not unique, but for any prime ideal $\mathfrak p\subset R$ the number of times $\mathfrak p$ occurs among the $\mathfrak p_i$ does not depend on the series thanks to Jordan-Hölder. This "multiplicity" of prime ideals has a direct geometrical meaning in the algebraic geometric case: for example, let $R$ be an integral domain and $M=R/I$ for some ideal $I\subset R$, then $\operatorname{Spec}M$ is a closed subscheme of the irreducible scheme $\operatorname{Spec}R$ (think of them as algebraic variety and subvariety when R is a finitely generated reduced algebra over an algebraically closed field); in this case, the prime ideals $\mathfrak p_i$ are precisely the ideals of the irreducible (and maybe embedded) components of $\operatorname{Spec}M$, or in other words the prime ideals associated to all primary ideals in the primary decomposition of $I$. So the number of times a $\mathfrak p$ appears among the $\mathfrak p_i$ can be thought of as the "multiplicity" of the corresponding component in the scheme, so that the sum of the multiplicities of its components is the length of the coordinate ring $M$ of the scheme $\operatorname{Spec}M$.

In general these multiplicities and lengths are used in algebraic geometry to count components of intersections (thus defining degrees of varieties with all its multitude of important consequences, like Bézout's theorem) and order of vanishing of rational functions to define principal divisors, so that one can arrive at linear and rational equivalence of divisors and cycles, forming fundamental algebraic invariant groups (Chow's) for varieties and schemes. For example, let $X$ be an algebraic variety with field of functions $K(X)$ and let $V\subset X$ be a subvariety of codimension 1 with local ring $\mathcal O_{X,V}$ (this can be defined as the localization of the regular ring at the generic point of $V$, seeing $X$ as a scheme). Then for any $f\in R^{\times}\subset K(X)$, the order of vanishing of $f$ at $V$ is defined to be the integer: $$\operatorname{ord}_V (f):=\operatorname{length}_{\mathcal O_{X,V}} \mathcal O_{X,V}/(f)\Rightarrow \\\text{ if } g=\frac{p}{q}\in K(X),\text{ with $p,q\in\mathcal O_{X,V}$ then }\operatorname{ord}_V (g):=\operatorname{ord}_V (p)-\operatorname{ord}_V (q),$$ which is well-defined precisely because the length of modules is additive on exact sequences thanks to Jordan-Hölder theorem again. Working with this kind of construction lets us define intersections for general cycles in schemes, generalizing the classical definition of intersection multiplicity of two projective plane curves $C^n,C^m\subset\mathbb{CP}^2$ defined locally by the zero loci of polynomials $f,g\in\mathbb C[x,y,z]$ of degrees $n,m$: let $P\in C^n\cap C^m$, then the intersection multiplicity at every such common point (i.e. the number of times the point must be counted) is $$m_P (C^n,C^m)=\operatorname{length}_{\mathbb C}(\mathcal O_{\mathbb P^2, P}/(f,g)).$$ Proving that the traditional intersection multiplicity, as defined by counting common solutions of the defining polynomials via elimination theory, is equal to the above length of modules (in this case = dimension of vector space), is the typical approach to show that traditional multiplicity is independent of the system of coordinates (that is why most algebraic geometry books nowadays define $m_P$ directly via lengths instead of algebraic root multiplicities via resultants of polynomials). Such multiplicities via lengths appear in Bézout's theorem and thus allow a definition of degree of subvarieties in those terms (sum of the lengths for all the corresponding quotient modules at every point of intersection of the subvariety with a generic linear subspace of complementary dimension).

Therefore Jordan-Hölder theorem for groups extends straightforwardly to modules and guarantees a well-defined intrinsic algebraic invariant which is a cornerstone in intersection theory for algebraic geometry.

Solution 2:

See definition of Hilbert function and its uses in Algebraic Geometry when it is applied on finitely generated $R$-modules. Look at the sequence $M=M_{n}\supseteq \dots\supseteq M_{i}\supseteq \dots\supseteq M_{0}=\{0\}$ where $\{M_{i}\}^{n}_{i=0}$ are submodules of $M$. It is a composition series if for each $1\leq i\leq n$ we have $\frac{M_{i}}{M_{i-1}}$ a simple $R$-module. Then we define $l(M):=$length of a composition series for the $R$-module $M$. (Note that because $M$ is finitely generated so we have such series for it.)

Definition: Let $M$ be graded module. Then

$H:\mathbb{Z}\rightarrow\mathbb{Z}$

$H(n)=l(M_{n})=\dim_{K}M_{n}<\infty$.

It has lots of uses in Algebraic Geometry. You can see some of them and then if you are interest you can try to make some equivalent for them in Group-Theory. A nice reference is "Multiplicity and Chern Class of Varieties" written by Roberts.

Also pay attention that every $\mathbb{Z}$-module can give us an Abelian group. From definition of modules like $M$ over ring $R$ with operators $+$ and $\cdot$, $M$ has to be an Abelian group with $+$. Conversely, one can check that if $G$ be an abelian group with operator $+$, by defining scaler multiplication $\cdot\; :\; \mathbb{Z}\times M\rightarrow M$ such that $k\cdot g:=g^{k}$ the additive power of $g$ in group structure of $G$, $G$ is a $\mathbb{Z}$-module.