Definition 11

Let G be a group with operators; the length of G is the upper bound of the integers $n$ such that there exists a strictly decreasing composition series of G $(G_i)_{0\leq i\leq n}$.

If G admits a Jordan-Hölder series, the length of G is the number of successive quotients of this series, as follows from the Corollary to Theorem 6. If G does not admit a Jordan-Hölder series, its length is infinite; by Proposition 9, for every strictly decreasing series of G there exists a strictly finer strictly decreasing series. The group consisting of the identity element is the only group with operators of length zero. A group with operators is simple if and only if it is of length 1.

Let G be a group with operators, H a normal stable subgroup of G, K the quotient G/H and $\pi:G\to K$ the canonical homomorphism. Let

be a decomposition series of H and

be a composition series of K. Writing $G_i=\pi^{-1}(K_i)$ for $0\leq i\leq p$ and $G_i=H_{i-p}$ for $p\leq i\leq n+p$, a composition series

of G is obtained. The sequence of quotients of $\Sigma$ is obtained by juxtaposing the sequence of quotients of $\Sigma''$ and the sequence of quotients of $\Sigma'$. If $\Sigma'$ and $\Sigma''$ are Jordan-Hölder series, $\Sigma$ is a Jordan-Hölder series of G, by Proposition 9. If H or K admits composition series of arbitrary length, so does G. We have proved: