The theory of powerful p-groups was created by A. Lubotzky and A. Mann [1987]; it was also anticipated in an earlier work of M. Lazard [1965]. Powerful p-groups have already found several applications in the theories of finite p-groups, of pro-p-groups, of residually finite groups, of groups with bounded ranks, of groups of given coclass, etc. One can say that the theory of powerful p-groups reflects the properties of the “linear part” of a finite p-group of given rank. Applications to finite p-groups with almost regular p-automorphisms are based on the bounds for the ranks in terms of the number of fixed points and the order of the automorphism (§2.2). The exposition in this chapter follows [A. Lubotzky and A. Mann, 1987] and includes some lemmas from [A. Shalev, 1993a] and [J. D. Dixon et al., 1991]. The proofs, however, are here inflated to a more verbose form, to make them accessible for a beginner; some sharper bounds are sacrificed for the same reasons. We shall consider only the case when p is an odd prime; the same results hold for p = 2, but the definitions and some proofs are a little different (although not more difficult) and are left as exercises to the reader.

Definitions and basic properties

Throughout the chapter, p denotes a fixed prime number, which is assumed odd, if not otherwise stated.