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:

  1. 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;
  2. f(z)=z2 (f: C* → C* is a homomorphism from the multiplicative group C* to itself) is again an epimorphism but not a monomorphism.

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