# 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 $G$:

- $\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$

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

### Factors

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 **factor groups** of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$:

- $\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$

are the quotient groups:

- $G_1 / G_0, G_2 / G_1, \ldots, G_i / G_{i - 1}, \ldots, G_n / G_{n-1}$

### Normal Series as Sequence of Homomorphisms

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$:

- $\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$

whose factor groups are:

- $H_1 = G_1 / G_0, H_2 = G_2 / G_1, \ldots, H_i = G_i / G_{i - 1}, \ldots, H_n = G_n / G_{n - 1}$

By Kernel of Group Homomorphism Corresponds with Normal Subgroup of Domain, such a series can also be expressed as a sequence $\phi_1, \ldots, \phi_n$ of group homomorphisms:

- $\set e \stackrel {\phi_1} {\to} H_1 \stackrel {\phi_2} {\to} H_2 \stackrel {\phi_3} {\to} \cdots \stackrel {\phi_n} {\to} H_n$

### Infinite Normal Series

A normal series may or may not terminate at either end:

- $\cdots \stackrel {\phi_{i - 1} } {\longrightarrow} H_{i - 1} \stackrel {\phi_i} {\longrightarrow} H_i \stackrel {\phi_{i + 1} } {\longrightarrow} H_{i + 1} \stackrel {\phi_{i + 2} } {\longrightarrow} \cdots$

Such a series is referred to as an **infinite normal series**.

The context will determine which end, if either, it terminates.

### Length

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.

## 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 not 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** $\sequence {G_i}_{i \mathop \in \set {0, 1, \ldots, n} }$ 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

- Results about
**normal series**can be found here.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 71$