Let $(G,\ast)$ and $(K,\star)$ be two groups. A group homomorphism is a function $\phi\colon G \to K$ such that $\phi (s \ast t) = \phi(s) \star \phi(t)$ for all $s,t \in G$

A composition of group homomorphisms is again a homomorphism.

Let $\phi\colon G\to K$ a group homomorphism. Then the kernel of $\phi$ is a normal subgroup of $G$ and the image of $\phi$ is a subgroup of $K$ Also, $\phi(g^n)=\phi(g)^n$ for all $g\in G$ and for all $n \in\Z$ In particular, taking $n=-1$ we have $\phi(g^{-1})=\phi(g)^{-1}$ for all $g \in G$ and taking $n=0$ we have $\phi(1_G)=1_K$ where $1_G$ and $1_K$ are the identity elements of $G$ and $K$ respectively.

Some special homomorphisms have special names. If the homomorphism $\phi\colon G\to K$ is injective, we say that $\phi$ is a monomorphism, and if $\phi$ is surjective we call it an epimorphism. When $\phi$ is both injective and surjective (that is, bijective) we call it an isomorphism. In the latter case we also say that $G$ and $K$ are isomorphic, meaning they are basically the same group (have the same structure). A homomorphism from $G$ on itself is called an endomorphism, and if it is bijective then it is called an automorphism.