# 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

• Results about kernels of group homomorphisms can be found here.