\begin{definition}[Definition:Kernel of Group Homomorphism]
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.
The '''kernel''' of $\phi$ is the subset of the domain of $\phi$ defined as:
:$\map \ker \phi := \phi^{-1} \sqbrk {e_H} = \set {x \in G: \map \phi x = e_H}$
where $e_H$ is the identity of $H$.
That is, $\map \ker \phi$ is the subset of $G$ that maps to the identity of $H$.
\end{definition}