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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.

The quotient map of groups sends each group element to its coset modulo a normal subgroup. It is a surjective group homomorphism, with group kernel exactly $N$. In the additive six-element example, it sends $0,3$ to $N$, $1,4$ to $1+N$, and $2,5$ to $2+N$. The quotient map of groups records the identification that produces the new group.

Let $N$ be a normal subgroup of $G$, and let $\pi :G\to G/N$ be the quotient map of groups. If a group homomorphism $\phi :G\to H$ has group kernel containing $N$, there is exactly one group homomorphism $\bar{\phi }:G/N\to H$ with $\phi =\bar{\phi }\circ \pi$. Its value is forced: $\bar{\phi }(gN)=\phi (g)$. Every measurement by a group homomorphism that ignores $N$ can thus be performed on the quotient group directly. This is the universal property of the construction.