Proposition 9

For a decomposition series of G to be a Jordan-Hölder series it is necessary and sufficient that all the quotients of the series be simple.

A decomposition series is strictly decreasing if and only if none of its successive quotients is reduced to the identity element. If a strictly decreasing composition series $\Sigma$ is not a Jordan-Hölder series, there exists a strictly decreasing composition series $\Sigma'$ which is finer than $\Sigma$ and distinct from $\Sigma$. There are therefore two consecutive terms $G_i, G_{i+1}$ of $\Sigma$ which are not consecutive in $\Sigma'$; let $H$ be the first term which follows $G_i$ in $\Sigma'$; $H$ is a normal stable subgroup of $G_i$, containing $G_{i+1}$ and distinct from the latter; hence $H/G_{i+1}$ is a normal stable subgroup of $G_i/G_{i+1}$, distinct from the latter and from the identity element; therefore $G_i/G_{i+1}$ is not simple. Conversely, if $\Sigma$ is a strictly decreasing composition series one of whose quotients $G_i/G_{i+1}$ is not simple, this quotient contains a normal stable subgroup other than itself and $\{e\}$, whose inverse image in $G_i$ is a normal stable subgroup $H$ of $G_i$, distinct from $G_i$ and $G_{i+1}$ (Theorem 4); it suffices to insert $H$ between $G_i$ and $G_{i+1}$ to obtain a strictly decreasing composition series distinct from $\Sigma$ and finer than $\Sigma$.