Definition:Kernel of Group Homomorphism

Definition
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 $H$.

Also denoted as
The notation $\map {\mathrm {Ker} } \phi$ can sometimes be seen.

It can also be presented as $\ker \phi$ or $\operatorname {Ker} \phi$, that is, without the parenthesis indicating a mapping.

Also see

 * Identity is in Kernel of Group Homomorphism where it is shown that $e_G \in \map \ker \phi$ where $e_G$ is the identity of $G$.


 * Kernel is Normal Subgroup of Domain