Composition length of extension group is sum of composition lengths

Definition
Suppose $$G$$ is a group, $$N$$ is a normal subgroup, and $$G/N$$ is the corresponding quotient group.

The composition length of $$G$$ is the sum of the composition length of $$N$$ and the composition length of $$G/N$$.

In particular, if $$G$$ is a group of finite composition length, then so are $$N$$ and $$G/N$$. Conversely, if both $$N$$ and $$G/N$$ have finite composition length, so does $$G$$.

Related facts

 * Chief length of extension group is bounded by sum of chief lengths
 * Composition length of direct product is sum of composition lengths
 * Chief length of direct product is sum of chief lengths