Most of this book is about pro-p groups. The purpose of this introductory chapter is to explain what these are and where they come from, and to establish their basic properties. Many of these properties belong to the larger class of profinite groups, and we begin by discussing these.

Notation If G is a topological group and X is a subset of G, we write X̄ to denote the closure of X in G, and denote by 〈X〉 the subgroup of G generated as an abstract group by X. We write X ≤oG, X ⊲oG, X ≤c G, X ⊲c G to denote: X is an open subgroup, open normal subgroup, closed subgroup, closed normal subgroup of G, respectively.

Profinite groups

Definition A profinite group is a compact Hausdorff topological group whose open subgroups form a base for the neighbourhoods of the identity.

Thus a discrete group is profinite if and only if it is finite. Since in a topological group any subgroup containing a non-empty open set is itself open, we see that the second part of the definition comes down to: every open set containing 1 contains an open subgroup. There are several equivalent definitions, some of which are discussed in the exercises; the most important one is given in Proposition 1.3, below. First we list some elementary consequences of Definition 1.1.

PropositionLet G be a profinite group.

(i) Every open subgroup of G is closed, has finite index in G, and contains an open normal subgroup of G.