A homomorphism which is onto. If it's also a monomorphism, then it is called an isomorphism.

This needn't be a monomorphism (i.e. it needn't be one to one) -- so it needn't be an isomorphism. Examples:

- Defining X* as X\{0} for any field X, the epimorphisms
**sign**: **R*** → {-1, +1} (from the multiplicative group **R*** to the multiplicative group {-1, +1}) is an epimorphism but not a monomorphism;
*f(z)*=*z*^{2} (*f*: C* → C* is a homomorphism from the multiplicative group **C*** to itself) is again an epimorphism but not a monomorphism.