A map $f: G \rightarrow H$ from group $G$ to group $H$ is called isomorphism if it is bijective (one to one map) and preserves the group operation, i.e. $\forall x,y \in G: f(x \cdot y)=f(x) \cdot f(y)$. If such a map exists we say that groups $G$ and $H$ are isomorphic.