Corollary

Let G be a group with operators and $(G_i)_{0\leq i\leq n}$ a composition series of G. The length of G is the sum of the lengths of the $G_i/G_{i+1}$, $0\leq i\leq n-1$.

If G and $G'$ are isomorphic groups with operators and G admits a Jordan-Hölder series, so does $G'$ and the Jordan-Hölder series of G and $G'$ are equivalent. However, non-isomorphic groups can have equivalent Jordan-Hölder series; such is true for $\mathbf{Z}/4\mathbf{Z}$ and $(\mathbf{Z}/2\mathbf{Z})\times(\mathbf{Z}/2\mathbf{Z})$, cf. no. 10.