Definition 9

*A composition series of a group with operators G is a finite sequence $(G_i)_{0 \leq i \leq n}$ of stable subgroups of G, with $G_0 = G$ and $G_n = \{e\}$ and such that $G_{i+1}$ is a normal subgroup of $G_i$ for $0 \leq i \leq n-1$. The quotients $G_i/G_{i+1}$ are called the quotients of the series. A composition series $\Sigma'$ is said to be finer than a composition series $\Sigma$ if $\Sigma$ is a series taken from $\Sigma'$.

If $(G_i)_{0 \leq i \leq n}$ and $(H_j)_{0 \leq j \leq m}$ are respectively composition series of two groups with operators G and H, they are said to be equivalent if $m = n$ and there exists a permutation $\phi$ of the interval $\{0, n-1\}$ of $\mathbf{N}$, such that the groups with operators $G_i/G_{i+1}$ and $H_{\phi(i)}/H_{\phi(i)+1}$ are isomorphic for all $i$.

Note that in general a series taken from a composition series $(G_i)$ is not a composition series, since for $j > i + 1$, $G_j$ is not in general a normal subgroup of $G_i$.