In this chapter we restrict attention to finite p-groups. It turns out that the key to understanding the structure of analytic pro-p groups lies in the properties of a special class of finite groups.

Definition (i) A finite p-group G is powerful if p is odd and G/Gp is abelian, or p = 2 and G/G4 is abelian.

(ii) A subgroup N of a finite p-group G is powerfully embedded in G, written N p.e. G, if p is odd and [N, G] ≤ Np,or p = 2 and [N, G] ≤ N4.

Thus G is powerful if and only if G p.e. G; and if N p.e. G then N ⊲ G and N is powerful. When p is odd, G is powerful if and only if Gp = Φ(G). One should think of ‘powerful’ as a generalization of ‘abelian’. We shall see that powerful p-groups (and, later, pro-p groups) share many of the simple structural features of abelian groups.

Lemma.Let G be a finite p-group and let N, K and W be normal subgroups of G with N ≤ W.

1. (i) If N p.e. G then NK/K p.e. G/K.

2. (ii) If p is odd and K ≤ Np, or if p = 2 and K ≤ N4, then N p.e. G if and only if N/K p.e. G/K.

3. (iii) If N p.e. G and x ∈ G then 〈N, x〉 is powerful.