Until relatively recently, it was received wisdom that there was no point in trying to classify finite p-groups up to isomorphism: there are just too many of them. Received wisdom was flouted in 1980 by Charles Leedham-Green and Mike Newman, who formulated a series of conjectures that amount to no less than a programme for the classification of all finite groups of prime-power order.

Quite remarkably, not only did the conjectures turn out to be exactly correct, but they were proved, over a period of about ten years, by following the guidelines proposed in the original paper of Leedham-Green and Newman.

We shall not repeat the conjectures, nor relate the interesting history and prehistory of their ultimate success; some pointers to the relevant literature are provided in the Notes at the end of the chapter. Our aim here is to explain how pro-p groups come into the picture, and to give a self-contained account of that part of the theory that relates to pro-p groups. Apart from its original motivation in the classification of finite groups, this makes a beautiful chapter in the theory of pro-p groups of finite rank.

The primary invariant used in the classification is the coclass of a finite p-group: the coclass of a group G of order pn is n – c where c is the nilpotency class of G.