Normal subgroup

For a much more thorough and detailed treatment of normal subgroups, see normal subgroup at the group properties wiki

Definition

Definitions in tabular format

Six equivalent definitions of normality are listed below. Note that each of these definitions (except the first one, as noted) assumes that we already have a group and a subgroup. Thus, to prove normality using any of these definitions, we first need to check that we actually have a subgroup.

No.	Shorthand	A subgroup of a group is normal in it if...	A subgroup $H$ of a group $G$ is normal in $G$ if ...	Additional comments
1	Homomorphism kernel	it is the kernel of a homomorphism from the group.	there is a homomorphism $\varphi$ from $G$ to a group $K$ such that the kernel of $\varphi$ is precisely $H\!$. In other words, $\varphi (x)\!$ is the identity element of $K$ if and only if $x\in H$.	In this case, we do not need to separately check that $N$ is a subgroup since the kernel of a homomorphism is automatically a subgroup.
2	Inner automorphism invariance	it is invariant under all inner automorphisms.	for all $g\in G$, $gHg^{-1}\subseteq H$. More explicitly, for all $g\in G,h\in H$, we have $ghg^{-1}\in H$.	Thus, normality is the invariance property with respect to the property of an automorphism being inner. This definition also motivates the term invariant subgroup for normal subgroup (which was used earlier).
3	Equals conjugates	it equals each of its conjugates in the whole group.	for all $g$ in $G$, $gHg^{-1}=H$.	This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
4	Left/right cosets equal	its left cosets are the same as its right cosets (that is, it commutes with every element of the group).	for all $g$ in $G$, $gH=Hg$.	When we say $gH=Hg$, we only mean equality as sets. It is not necessary that $gh=hg$ for $h\in H$. That stronger condition defines central subgroup.
5	Union of conjugacy classes	it is a union of conjugacy classes.	$H$ is a union of conjugacy classes in $G$
6	Commutator inside	it contains its commutator with the whole group.	the commutator $[H,G]$ (which coincides with the commutator $[G,H]$) is contained in $H$.

Notation and terminology

For a subgroup $H\!$ of a group $G\!$, we denote the normality of $\!H$ in $\!G$ by $H{\underline {\triangleleft }}G$ or $G{\underline {\triangleright }}H$. In words, we say that $\!H$ is normal in $\!G$ or a normal subgroup of $\!G$.