Definition:Normal Subgroup

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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.



Historical Note

The conceptual importance of the normal subgroup was recognised by √Čvariste Galois.

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


Sources