Let $(G,\ast_G)$ and $(H,\ast_H)$ be \reftext{def:group-2026a}{groups} and let $\varphi:G\to H$ be a \reftext{def:group-homomorphism-isomorphism-2026a}{group homomorphism}. Then the \reftext{def:kernel-image-group-homomorphism-2026a}{kernel} of $\varphi$ is a \reftext{def:subgroup-2026a}{subgroup} of $G$, and the image of $\varphi$ is a subgroup of $H$.