Let $(G,\ast_G)$ and $(H,\ast_H)$ be \reftext{def:group-2026a}{groups} with identity elements $e_G$ and $e_H$ and inverses as in \ref{thm:group-identity-inverse-uniqueness-2026a}, and let $\varphi:G\to H$ be a \reftext{def:group-homomorphism-isomorphism-2026a}{group homomorphism}. Then

and for every $a\in G$

where on the left the inverse is taken in $G$ and on the right in $H$.