4 - Uniformly powerful groups

Summary

In the previous chapter, we saw that every pro-p group of finite rank has an open normal subgroup which is powerful. In this chapter we show that this subgroup may be chosen so as to satisfy a slightly stronger condition, that of being ‘uniformly powerful’. We then show that uniformly powerful groups have a remarkable property: the group operation can be ‘smoothed out’, to give a new, abelian, group structure, and this new abelian group is in a natural way a finitely generated free ℤ_p-module.

When, in Part II, we come to consider a pro-p group of finite rank as an analytic group, we shall see that this ℤ_p-module structure provides a natural co-ordinate system on the group. More immediately, we obtain, free of charge, a faithful linear representation for the automorphism group of any uniformly powerful pro-p group.

Uniform groups

Definition A pro-p group G is uniformly powerful if

1. (i) G is finitely generated,

2. (ii) G is powerful, and

3. (iii) for all i, |P_i(G) : P_i+1(G)| = |G : P₂(G)|.

We shall usually abbreviate ‘uniformly powerful’ to ‘uniform’.

If G is a pro-p group satisfying (i) and (ii) of this definition, then we know from Theorem 3.6 that the pth power map x ↦ x^p induces an epimorphism fi : P_i(G)/P_i+1(G) → P_i+1(G)/P_i+2(G), for each i; condition (iii) of Definition 4.1 is clearly equivalent to:

(iii)′ for each i ≥ 1, the map f_i is an isomorphism.