# Definition:Normal Subgroup

## Definition

Let $G$ be a group.

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

$N$ is a normal subgroup of $G$ iff:

### 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

$\forall g \in G: \left({n \in N \iff g \circ n \circ g^{-1} \in N}\right)$
$\forall g \in G: \left({n \in N \iff g^{-1} \circ n \circ g \in N}\right)$

### 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).

## 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:

$\forall g \in G: \left({n \in N \iff g \circ n \circ g^{-1} \in N}\right)$
$\forall g \in G: \left({n \in N \iff g^{-1} \circ n \circ g \in N}\right)$

which is another way of stating that $N$ is normal iff $N$ is invariant under all inner automorphisms of $G$.