Part III Profinite Groups Gareth Wilkes Lent Term 2020 Introduction Much of the story of pure mathematics can be expressed as a desire to answer the question "When are two objects different?". Showing that two objects are `the same' in some sense may be considerably easier than proving the contrary. For example, the theory of infinite cardinal numbers may be seen as answering the question `why is the set of real numbers different from the set of rational numbers?'. In this context it is considerably easier, for example, to exhibit a bijection between Z and Q than to show that no bijection between Q and R can possibly exist. In many cases we show that objects are different by defining `invariants': quantities which ideally may be computed, are preserved under isomorphism, and are easier to tell apart than the original object. Some examples of invariants you may have met are given below. • For a (finite-dimensional) vector space V , the base field F and the dimen- sion dimF (V ) determine V up to linear isomrphism. • For an algebraic field extension K=Q, the degree [K : Q] and, under some conditions, the Galois group of K over Q. • For a topological space X, the properties of compactness, connectedness, and path-connectedness may be considered to be invariants. • For a simplicial complex X, the homology groups H∗(X). • For a path-connected topological space X, the fundamental group π1X. This last example is a somewhat odd invariant. It is extremely powerful and often gives fine control over the space X. However, groups are in general quite as difficult to distinguish as spaces. In fact, to some extent, they are impossible to distinguish, even for groups that are given by finite presentations: it is even known (by the Adian-Rabin Theorem) that there can be no algorithm that takes as input a finite presentation and decides whether the presentation gives the trivial group. Never mind the harder question of an algorithm taking in two finite presentations and deciding if they give isomorphic groups. This theorem does not stop us from attempting to design algorithms that distinguish between groups under certain additional conditions. For example, on class of groups for which it is definitely possible to decide isomorphism is the class of finite groups: between any two finite groups there are only finitely many possible bijections, so one can just check all possible bijections to see whether they're isomorphisms. This would be a pretty terrible algorithm, but it would eventually decide the question. 1 2 For infinite finitely presented groups, this suggests one conceivable way to tell two groups apart. One can begin to list finite quotients of the two groups and compare the two lists. If these lists differ at some point (that is, there is a finite group which is a quotient of one but not the other), then we will have proved that the two groups are not isomorphic. Of course, we know that this procedure cannot always work|but it is still worthwhile to investigate when it does. For theoretical purposes, one does not really want to work with such an unwieldy object as `the set of all finite quotients' of a group G. Instead one combines these finite groups into one limiting object, the profinite completion of the group. Such a `limit of finite groups' is the central object of the course, a profinite group. Most of the time we will be studying profinite groups as interesting ob- jects in their own right, or as the profinite completion of a known group. It is worthwhile, however, to mention in the introductory section two other areas of mathematics where profinite groups arise naturally. Neither of these areas is a course pre-requisite, so we will not see these concepts further in this course and can be ignored for those without the precise necessary background. One key area in which profinite groups arise naturally (and indeed, in which they were first defined) is Galois theory. Specifically the Galois group of an infinite Galois extension is naturally a profinite group. Let us see one example. Consider the field extension K obtained from Q by adjoining all pn-th roots of unity for some fixed prime p, as n varies over N. Define also the extension KN n of Q by adjoining all p -th roots for n ≤ N. Then the KN form an increasing S union of subfields of K such that K = KN . Each extension K=KN is Galois, as are the extensions KN =Q, K=Q. It follows that there are natural quotients ∼ N × Gal(K=Q) ! Gal(KN =Q) = (Z=p Z) and in fact the Galois group Gal(K=Q) may be considered to be a `limit' of these finite groups in a certain sense (to be defined soon). This behaviour is typical: every Galois group of an infinite Galois extension may be considered to be a limit of finite Galois groups. I will also mention that profinite groups also appear in algebraic geometry in the guise of ´etalefundamental groups, though to be perfectly honest I'm not qualified to expand upon this remark further. The second part of this course also concerns a certain `invariant' intended to distinguish groups from one another, the theory of group cohomology. As the name may suggest, this theory is intimately connected with the homology theory of spaces as studied in Part II Algebraic Topology, but has its own indepedent strengths and weaknesses. Much like the homology group of a space, we will take a group G and define a collection of abelian groups depending on G, called the cohomology groups of G. These can be powerful theoretical tools in many contexts. As one key application we will show how cohomology groups provide a solution to the following natural question: Given a group G and an abelian group A, how many groups E can there be such that A/E with E=A = G? It is readily seen that a good answer to this question would allow one in principle to classify all groups with order pn, or more generally all solvable groups, so this 3 particular application of cohomology theory may be seen as carrying on some of the story from IB Group Theory. At the conclusion of the course we will combine our two main threads into the cohomology theory of profinite groups, and establish some remarkably strong theorems which show that in some ways our profinite groups are actually better behaved than normal groups! First however, we must lower our sights back down to fundamentals. We begin with some category theory. Chapter 1 Inverse limits 1.1 Categories and limits As was mentioned in the introductory section, we will be seeking to combine the information contained in, for example, the finite quotients of a given group into one object which we can study. Let us see some basic constructions which combine several objects into one. Let A and B be two sets, with no particular relationship between them. How might these be combined into one object? Two constructions should come to mind fairly quickly: the product A × B and the disjoint union A t B. What properties of A × B and A t B describe their relationship to A and B? The disjoint union comes equipped with inclusion maps iA : A ! A t B and iB : B ! A t B. Furthermore, it is in some sense the `minimal' object that sensibly contains both A and B: for any other set Z with maps jA : A ! Z and jB : B ! Z, there is a unique function f : A t B ! Z which restricts to jA and jB, defined simply by f(a) = jA(a) for a 2 A and f(b) = jB(b) for b 2 B. This situation would be represented diagrammatically as follows: A iA A t B iB B 9! f (1.1) jA jB Z This diagram commutes in the sense that `following the arrows' gives the same result whatever path is taken|for example, f ◦ iA = jA. The sort of property described above|where some data (e.g. the set Z and the functions jA and jB) implies the unique existence of something else (here, the map f) is called a universal property. We see that the disjoint union tells us about maps leaving A and B. What about maps to A and B? This is where the product comes in. The product A×B is equipped with the usual projection maps pA : A×B ! A and pB : A×B ! B. Once again, there is a `universality' condition too. Given a set Z and maps qA : Z ! A and qB : Z ! B, there is a unique map g : Z ! A × B such that pA ◦ g = qA and pB ◦ g = qB, defined by g(c) = (qA(c); qB(c)). In the form of a 4 CHAPTER 1. INVERSE LIMITS 5 diagram: Z q q A 9! g B (1.2) A A × B B pA pB Note the similarity between these two diagrams. We say that the product and disjoint union of sets are dual to one another. Duality in this situation is often denoted by the prefix `co-': so the disjoint union may also be called a coproduct. Let us now move back to group theory, and insist that all our functions are group homomorphisms. We can still ask what groups may play the roles of AtB and A × B in the diagrams above. The product A × B is of course a perfectly sensible group, and satisfies the same universal property: the projection maps pA and pB are group homomorphisms, and provided we assume Z is a group and qA and qB are homomorphisms, then so is g = (qA; qB).