Group homomorphism/Kernel/Definition

Kernel (group homomorphism)

Let ${}G$ and ${}H$ be groups, and let

$\varphi \colon G\longrightarrow H$

be a group homomorphism. Then the preimage of the neutral element is called the kernel of ${}\varphi$, denoted by

${}\operatorname {kern} \varphi =\varphi ^{-1}(e_{H})={\left\{g\in G\mid \varphi (g)=e_{H}\right\}}\,.$