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A subgroup $N\le G$ is a normal subgroup when $gN{g}^{-1}=N$ 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 $N⊴G$.

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 $g=(123)$. Thenwhich 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.

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.

For a subgroup $N\le H$ and $h\in H$, a left coset is $hN=\{hn:n\in N\}$ and a right coset is $Nh=\{nh:n\in N\}$. A normal subgroup satisfies $hN{h}^{-1}=N$ for every $h\in H$, equivalently $hN=Nh$.

If $N$ is normal, the quotient group is the set $H/N$ of cosets, with multiplicationTo check independence of representatives, replace $h,k$ by $h{n}_1,k{n}_2$. Thenbecause normality puts $k^{-1}{n}_{1}k$ in $N$. Thus the operation is well defined. Associativity follows from that in $H$, the identity is $N$, and the inverse of $hN$ is $h^{-1}N$. This proves the group axioms for $H/N$ and explains the role of coset multiplication is well-defined exactly for normal subgroups.