Jordan--Hölder theorem

Let \(G\) be a group, and let \[ G=G_0 \trianglerighteq G_1 \trianglerighteq \cdots \trianglerighteq G_r=\theset{e} \quad\text{and}\quad G=H_0 \trianglerighteq H_1 \trianglerighteq \cdots \trianglerighteq H_s=\theset{e} \] be two composition series of \(G\). Then \(r=s\), and there is a permutation \(\pi\) of \(\theset{1,\ldots,r}\) such that \(G_{i-1}/G_i\cong H_{\pi(i)-1}/H_{\pi(i)}\) for every \(i\).