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

\(\ds \forall g \in G: \, \) \(\ds g \circ N \circ g^{-1}\) \(\subseteq\) \(\ds N\)
\(\ds \forall g \in G: \, \) \(\ds g^{-1} \circ N \circ g\) \(\subseteq\) \(\ds N\)


Definition 4

\(\ds \forall g \in G: \, \) \(\ds N\) \(\subseteq\) \(\ds g \circ N \circ g^{-1}\)
\(\ds \forall g \in G: \, \) \(\ds N\) \(\subseteq\) \(\ds g^{-1} \circ N \circ g\)


Definition 5

\(\ds \forall g \in G: \, \) \(\ds N\) \(=\) \(\ds g \circ N \circ g^{-1}\)
\(\ds \forall g \in G: \, \) \(\ds N\) \(=\) \(\ds g^{-1} \circ N \circ g\)


Definition 6

\(\ds \forall g \in G: \, \) \(\ds \leftparen {n \in N}\) \(\iff\) \(\ds \rightparen {g \circ n \circ g^{-1} \in N}\)
\(\ds \forall g \in G: \, \) \(\ds \leftparen {n \in N}\) \(\iff\) \(\ds \rightparen {g^{-1} \circ n \circ g \in N}\)


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:

\(\ds \forall g \in G: \, \) \(\ds \leftparen {n \in N}\) \(\iff\) \(\ds \rightparen {g \circ n \circ g^{-1} \in N}\)
\(\ds \forall g \in G: \, \) \(\ds \leftparen {n \in N}\) \(\iff\) \(\ds \rightparen {g^{-1} \circ n \circ g \in N}\)


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

See Inner Automorphism Maps Subgroup to Itself iff Normal.


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:

\(\ds \) \(\) \(\ds S_3\)
\(\ds \) \(\) \(\ds \set e\)
\(\ds \) \(\) \(\ds \set {e, \tuple {123}, \tuple {132} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {12} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {13} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {23} }\)


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 Évariste Galois.

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


Sources

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