In general, left cosets need not coincide with right cosets. If they do, we have a case that deserves special attention. Let G be a group.

Theorem

Let H be a subgroup of G. The following assertions are equivalent.

  1. gH = Hg for every g G.
  2. ghg-1 H for every g, h G.

If H satisfies these properties, it is called a normal subgroup of G.

The following result shows essentially how they appear.

Proposition

If f : G -> G' is a morphism of groups, then Ker(f) is a normal subgroup of G.

Later we shall see that every normal subgroup is the kernel of a morphism.