Definition:Normal Series/Length

Definition
Let $G$ be a group whose identity is $e$. Let $\left \langle {G_i}\right \rangle_{i \in \left[{0 \,.\,.\, n}\right]}$ be a normal series for $G$:
 * $\left \langle {G_i}\right \rangle_{i \in \left[{0 \,.\,.\, n}\right]} = \left({\left\{{e}\right\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_{n-1} \triangleleft G_n = G}\right)$

The length of $\left \langle {G_i}\right \rangle_{i \in \left[{0 \,.\,.\, n}\right]}$ is the number of (normal) subgroups which make it.

In this context, the length of $\left \langle {G_i}\right \rangle_{i \in \left[{0 \,.\,.\, n}\right]}$ is $n$.

If such a normal series is infinite, then its length is not defined.