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 $G$:
 * $\left\{{e}\right\} = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$

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

Such a series may or may not terminate at either end w

Also known as
A normal series is also referred to as:


 * A normal tower
 * A subinvariant series
 * A sequence of groups

Also defined as
Note that from Normality Relation is Non-Transitive, it is not necessarily the case that if $G_a \lhd G_b \lhd G_c$ then $G_a \lhd G_c$.

Consequently, some sources specify that all subgroups $G_i$ in a normal series $\left \langle {G_i}\right \rangle_{i \in \left[{0 \,.\,.\, n}\right]}$ be normal subgroups of $G$ itself, as well as being normal subgroups of the next in sequence $G_{i+1}$.

Such a sequence in which it is not necessarily the case where $G_i$ is normal in $G$ for all $i$ is, in such a context, referred to as a subnormal series.

Also see

 * Definition:Subnormal Subgroup


 * Definition:Composition Series