TITLE: Applications of the Jordan-Hölder Theorem.
QUESTION [29 upvotes]: All books about group theory bring the Jordan-Hölder theorem, but does not give applications. I only saw in Rotman's book the Fundamental Theorem of Arithmetic being proved as a consequence of this theorem.
What is a nice application of this theorem? Any references?

REPLY [3 votes]: 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.