Let $(G,\ast_G)$ and $(H,\ast_H)$ be \reftext{def:group-2026a}{groups}. A function $\varphi:G\to H$ is called a \textbf{group homomorphism} if

A group homomorphism $\varphi:G\to H$ is called:

1. a \textbf{monomorphism} if it is injective, that is, if $\varphi(a)=\varphi(b)$ implies $a=b$ for all $a,b\in G$;
2. an \textbf{isomorphism} if it is a \reftext{def:bijection-sets-2026a}{bijection} from $G$ onto $H$.

The groups $(G,\ast_G)$ and $(H,\ast_H)$ are called \textbf{isomorphic} if there exists an isomorphism from $G$ onto $H$.

When the two operations are clear from the context we suppress them and write the defining condition as $\varphi(ab)=\varphi(a)\varphi(b)$.