Constructing finite p-groups

Abstract

Abstract This chapter begins the process of constructing p-groups of coclass r as given by Leedham-Green (1994b). A class of ߢconstructibleߣ p-groups is defined, and the structure theorem will say that every p-group P of coclass r has a normal subgroup N, of order bounded in terms of p and r, such that P/N is constructible. Recall that it has already been shown in Chapter 6 that a p-group of coclass r has a subgroup of nilpotency class 2 and small index. A constructible p-group is obtained from a finite homomorphic image of a p-adic space group of finite coclass by Twisting' the image of the translation subgroup, thus making it of nilpotency class 2. Section 8.1 presents the basic theory involved in carrying out this ‘twisting’ operation in the context of central extensions of modules for a group. The structure theorem will be proved in Chapter 11 after p-adic space groups have been investigated in some detail in Chapter 10.