Given two groups N and G, the task of extension theory is to determine all extensions of N by G: that is, groups E having N as a normal subgroup and E/N ≅ G. This is a highly developed subject, but its general recipes are difficult to apply in a really illuminating way when dealing with infinite groups. An exception is when the theory tells us that the extension necessarily splits, i.e. that E has a subgroup H with N∩ H = 1 and NH = E (so that H ≅ E/N ≅ G); such an H is called a complement for N in E. After some necessary preparations in section A, section B gives a splitting result of this kind for the case where N is abelian, G is nilpotent, and both are finitely generated.

Suppose E is a polycyclic group and N = Fitt (E). We know from Chapter 2 that then G = E/N is abelian-by-finite; taking nilpotent groups and abelian-by-finite groups as fairly well understood, it would be very satisfactory if there was a straightforward way to describe all extensions of such an N by such a G. Unfortunately (or perhaps fortunately, for the interest of the subject), the answer given by general extension theory is too complicated to be of much practical use; in particular we shall see in the final chapter that such an extension can be very far from having to split.