Let $(G,\ast_G)$ and $(H,\ast_H)$ be \reftext{def:group-2026a}{groups}, let $e_H$ be the identity element of $H$ as provided by \ref{thm:group-identity-inverse-uniqueness-2026a}, and let $\varphi:G\to H$ be a \reftext{def:group-homomorphism-isomorphism-2026a}{group homomorphism}.

The \textbf{kernel} of $\varphi$ is the subset of $G$ given by

and the \textbf{image} of $\varphi$ is the subset of $H$ given by