We have seen that to every homomorphism we can associate two distinguished subgroups: and . It is natural to ask a converse question: given a subgroup of a group , is this subgroup the kernel or the image of some homomorphism?

It is easy that every subgroup is the image of some homomorphism. Indeed, if , define by . Then it is clear that .

The situation for kernels is different. In order to describe it, we introduce a special class of subgroups.

We now return to our investigation of the kernels.

Edmund F Robertson

11 September 2006