After studying kernels, we have seen that every kernel is a normal subgroup. The next statement goes in the reverse direction: every normal subgroup occurs as a kernel. The focus keyword natural homomorphism refers to the canonical map from a group onto a quotient group. If $H\trianglelefteq G$, the quotient group $G/H$ consists of cosets of $H$, and the natural homomorphism sends an element $a$ to the coset $aH$. This map is simple, but it is one of the main bridges between homomorphisms and quotient groups. In this lesson we prove that the map is a homomorphism and compute its kernel exactly.

Let $G$ be a group and let $H\trianglelefteq G$. The \textbf{natural homomorphism} from $G$ onto $G/H$ is the function $\pi:G\to G/H$ defined by $$\begin{equation*} \pi(a)=aH \end{equation*}$$ for every $a\in G$.