Let G and G' be two groups. By f(G) we denote the image of f, so

f(G) = {f(x) | x G}.

By Ker(f) we denote the inverse image of e', also called kernel of f; that is

Ker(f) = {x G | f(x) = e'}.

Part 3 below shows its importance, which is very similar and actually a generalization of the vector space case.

Theorem 

Let f : G -> G' be a morphism.

  1. The image f(G) is a subgroup of G'.
  2. The kernel Ker(f) is a subgroup of G.
  3. The morphism f is injective if and only if Ker(f) = {e}.
  4. If f is an isomorphism, then so is its inverse, the map f-1 : G' -> G.

Later we shall see that kernels are a special kind of subgroup.