# Definition:Normal Subgroup

## Contents

## Definition

Let $G$ be a group.

Let $N$ be a subgroup of $G$.

$N$ is a **normal subgroup of $G$** if and only if:

### Definition 1

- $\forall g \in G: g \circ N = N \circ g$

### Definition 2

- Every right coset of $N$ in $G$ is a left coset

that is:

- The right coset space of $N$ in $G$ equals its left coset space.

### Definition 3

- $\forall g \in G: g \circ N \circ g^{-1} \subseteq N$
- $\forall g \in G: g^{-1} \circ N \circ g \subseteq N$

### Definition 4

- $\forall g \in G: N \subseteq g \circ N \circ g^{-1}$
- $\forall g \in G: N \subseteq g^{-1} \circ N \circ g$

### Definition 5

- $\forall g \in G: g \circ N \circ g^{-1} = N$
- $\forall g \in G: g^{-1} \circ N \circ g = N$

### Definition 6

- $\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$
- $\forall g \in G: \paren {n \in N \iff g^{-1} \circ n \circ g \in N}$

### Definition 7

- $N$ is a normal subset of $G$.

## Notation

The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \lhd G$.

A normal subgroup is often represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal).

To use the notation introduced in the definition of the conjugate:

- $N \lhd G \iff \forall g \in G: N^g = N$

## Also known as

It is usual to describe a **normal subgroup of $G$** as **normal in $G$**.

Some sources refer to a **normal subgroup** as an **invariant subgroup** or a **self-conjugate subgroup**.

This arises from Definition 6:

- $\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$
- $\forall g \in G: \paren {n \in N \iff g^{-1} \circ n \circ g \in N}$

which is another way of stating that $N$ is **normal** if and only if $N$ stays the same under all inner automorphisms of $G$.

Some sources use **distinguished subgroup**.

## Examples

### Normal Subgroups of Symmetric Group on $3$ Letters

Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:

- $\begin{array}{c|cccccc} \circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$

Consider the subgroups of $S_3$:

The subsets of $S_3$ which form subgroups of $S_3$ are:

\(\displaystyle \) | \(\) | \(\displaystyle S_3\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set e\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set {e, \tuple {123}, \tuple {132} }\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set {e, \tuple {12} }\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set {e, \tuple {13} }\) | |||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle \set {e, \tuple {23} }\) |

Of those, the normal subgroups in $S_3$ are:

- $S_3, \set e, \set {e, \tuple {123}, \tuple {132} }$

## Also see

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

- Definition:Subnormal Subgroup
- Definition:Abnormal Subgroup
- Definition:Weakly Abnormal Subgroup
- Definition:Contranormal Subgroup
- Definition:Self-Normalizing Subgroup
- Definition:Pronormal Subgroup
- Definition:Weakly Pronormal Subgroup
- Definition:Paranormal Subgroup
- Definition:Polynormal Subgroup

## Historical Note

The conceptual importance of the **normal subgroup** was recognised by Évariste Galois.

He discussed it in the letter he wrote to his friend **Auguste Chevalier** the night before the duel that killed him.

## Sources

- 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.7$