After proving the basic properties of homomorphisms, the next natural question is which elements disappear under a homomorphism. The focus keyword kernel of a homomorphism refers to the set of domain elements whose image is the identity element of the codomain. The kernel measures the amount of collapse caused by the map. If the kernel is only the identity, the homomorphism keeps different elements separated. If the kernel contains more than the identity, several elements of the domain become indistinguishable in the codomain. In this lesson we define the kernel, prove that it is a normal subgroup, and use it to test one-to-one homomorphisms.

Let $f:G\to G_1$ be a homomorphism of groups, and let $e_1$ be the identity element of $G_1$. The \textbf{kernel of the homomorphism} $f$ is the set $$\begin{equation*} \ker f=\{a\in G\mid f(a)=e_1\}. \end{equation*}$$ Equivalently, $\ker f=f^{-1}(\{e_1\})$.