Group Homomorphism and Isomorphism

Let $(G,\ast_G)$ and $(H,\ast_H)$ be groups. A function $\varphi:G\to H$ is called a group homomorphism if

$$\varphi(a\ast_G b)=\varphi(a)\ast_H\varphi(b)\qquad\text{for all } a,b\in G.$$

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

1. a monomorphism if it is injective, that is, if $\varphi(a)=\varphi(b)$ implies $a=b$ for all $a,b\in G$;
2. an isomorphism if it is a bijection from $G$ onto $H$.

The groups $(G,\ast_G)$ and $(H,\ast_H)$ are called 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 $\varphi(ab)=\varphi(a)\varphi(b)$.