Definition:Normal Series

Definition
Let $G$ be a group whose identity is $e$.

A normal series for $G$ is a sequence of (normal) subgroups of $G$:
 * $\left\{{e}\right\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$

where $G_{i-1} \triangleleft G_i$ denotes that $G_{i-1}$ is a proper normal subgroup of $G_i$.

Factors
The factor groups (or just factors) of a normal series:
 * $\left\{{e}\right\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$

are the quotient groups:
 * $G_0 / G_1, G_1 / G_2, \ldots, G_{i-1} / G_i, \ldots, G_{n-1} / G_n$

Length
The length of a normal series is the number of (normal) subgroups which make it.

Alternative Names
A normal series is also known as:


 * A subnormal series
 * A normal tower
 * A subinvariant series