Normal subgroup
= Normal subgroup
{title2=$N\trianglelefteq G$}
{wiki}
A <subgroup> $N\leq G$ is a <normal subgroup> when $gNg^{-1}=N$ for every $g\in G$. In other words, <conjugation> by any <group element> preserves the <subgroup> as a <set>, although it may exchange its individual elements. Equivalently, $gN=Ng$ for every $g$: the left and right <cosets> of $N$ agree. Write $N\trianglelefteq G$.
Every <subgroup> of an <abelian group> is a <normal subgroup>, because <conjugation> fixes each element. Normality can also hold in a nonabelian <group>: in $S_3$, the three even <permutations> form the <alternating group> $A_3=\{e,(123),(132)\}$, a <normal subgroup> as the <group kernel> of the <sign of a permutation>. Its two <cosets> separate the even and odd <permutations>.
In contrast, $H=\{e,(12)\}$ is a <subgroup> of $S_3$ but is not a <normal subgroup>. With <function composition> of <permutations> acting rightmost first, choose $g=(123)$. Then
$$
gH=\{(123),(13)\},\qquad Hg=\{(123),(23)\},
$$
which are different <cosets>. Also $g(12)g^{-1}=(23)\notin H$, directly contradicting invariance under <conjugation>. Thus a <subgroup> alone does not suffice for <quotient group> multiplication.
\Image[/normal-subgroup-coset-contrast.png]
{title=A <normal subgroup> gives matching left and right <cosets>, while a nonnormal <subgroup> can give different ones}
The practical value of normality is <coset multiplication is well-defined exactly for normal subgroups>. Each <coset> can act as one element of a <quotient group>, with results independent of representatives. Conversely every <normal subgroup> is the <group kernel> of its <quotient map of groups>, so normality also identifies which distinctions a <group homomorphism> can consistently forget.