Definition:Normal Series/Length
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Definition
Let $G$ be a group whose identity is $e$.
Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$:
- $\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n-1} \lhd G_n = G}$
The length of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ is the number of (normal) subgroups which make it.
In this context, the length of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ is $n$.
If such a normal series is infinite, then its length is not defined.