Once a function has passed the homomorphism test, we can ask whether it is onto, one-to-one, or both. The focus keyword epimorphism monomorphism isomorphism automorphism collects the four standard names used for these special cases. A homomorphism may preserve the operation while still losing information or missing part of the codomain. A monomorphism preserves distinctness, an epimorphism reaches every codomain element, and an isomorphism does both. An automorphism is an isomorphism from a group to itself. These words are not decorative vocabulary; they describe exactly how much structure the map preserves and how completely it represents one group inside another.

Let $f:G\to G_1$ be a homomorphism. Then $f$ is called an \textbf{epimorphism} if $f$ is onto. Equivalently, $$\begin{equation*} f(G)=G_1. \end{equation*}$$