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A Crash Course on Group Theory

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A Crash Course on Group Theory A Crash Course on Group Theory Peter J. Cameron School of Mathematics and Statistics University of St Andrews North Haugh St Andrews, Fife KY16 9SS UK [email protected] ii Contents 1 Finite groups 1 1.1 What is a group? . .1 1.2 Subgroups . .6 1.3 Group actions . .9 1.4 Homomorphisms and normal subgroups . 11 1.5 Examples of groups . 14 1.6 Sylow’s theorem . 21 1.7 The Jordan–Holder¨ theorem . 23 1.8 The finite simple groups . 25 1.9 Groups of prime power order . 27 2 Infinite groups 29 2.1 Free groups . 29 2.2 Presentations of groups . 32 2.3 Algorithmic questions . 34 2.4 Periodic and locally finite groups . 37 2.5 Residually finite and profinite groups . 39 2.6 Oligomorphic permutation groups . 41 iii iv CONTENTS Preface On a visit to Universidade de Lisboa in November 2016, I was asked to give a “crash course” in group theory. The only specifications were that the course should cover both finite and infinite groups and should be accessible to students. This is a tall order. I have tried to meet it by starting at the beginning, moving fairly fast, omitting many proofs (this means leaving many proofs to the reader). But I hope the result is still of some use, so I am making the notes of the course available. There is a clear focus in the chapter on finite groups: we want to be able to describe them all. The Jordan–Holder¨ theorem reduces the problem to describing the finite simple groups and how general groups are built out of simple groups: the Classification of Finite Simple Groups solves the first part; a discussion of groups of prime power order shows that we cannot expect a nice solution to the second. For infinite groups, such a focus is much more difficult to obtain. There is no general theory of infinite groups, and group theorists have imposed various finite- ness conditions on their groups. I discuss, somewhat in the manner of a tourist guide, free groups, presentations of groups, periodic and locally finite groups, residually finite and profinite groups, and my own interest, oligomorphic permu- tation groups. I thank the students for their interest and their questions. v CHAPTER 1 Finite groups The theory of finite groups has seen a huge breakthrough in the last half-century: the Classification of Finite Simple Groups. My aim in this chapter is to introduce group theory, and to develop enough of the theory of finite groups that you can understand why this classification is important and the kind of problems to which it has been applied. 1.1 What is a group? By this question, I mean not just what the definition of a group is, but how we might think about groups. By way of illustration, in Alexander Masters’ biography of Simon Norton, The Genius in my Basement, the biographer thinks that the birth of group theory was the moment when the axioms for a group were first written down, and is horrified to find that his subject hasn’t the faintest idea of when this happened or who wrote the axioms down!1 In fact, at that point, group theory was a fairly mature subject; it had existed for a century, and some major theorems had been proved. So in what sense was this activity “group theory”? I’ll distingish three types of object which have been regarded as groups. (This is not a strict historical account; rather, it is to show where our present definition 1It was Walther von Dyck, in 1882. 1 2 Chapter 1. Finite groups came from.) Symmetry groups. The symmetry group G of a mathematical object O is the set of all transformations of O which preserve its structure. Lewis Carroll wrote: You boil it in sawdust: you salt it in glue: You condense it with locusts and tape: Still keeping one principal object in view – To preserve its symmetrical shape. Note that (a) every element of G is a bijection; (b) the composition of symmetries is a symmetry; (c) the identity transformation, which leaves everything where it is, is a sym- metry; (d) the inverse transformation of every symmetry is a symmmetry. Exercise: Take some familiar objects (a regular polygon, the Euclidean plane), and describe the symmetries. Transformation groups (or permutation groups). Let W be a set. A transfor- mation group on W is a set G of transformations of W (maps W ! W) satisfying (a) G is closed under composition; (b) the identity transformation belongs to G; (c) G is closed under taking inverse transformations. (Implicit in the third condition is that the elements of G are bijections; otherwise they would not have inverses.) Clearly any symmetry group is a transformation group. The converse is also true: this means that, given any transformation group G, we can find an object O of some kind such that G is the symmetry group of O. The construction is not enlightening. However, there is a good question here, which has been much investigated: 1.1. What is a group? 3 Question: Given a type of mathematical object (graph, partial order, field), which transformation groups are symmetry groups of an object of this type? Abstract groups. Now we come to the modern definition. A group G is a set of elements with a binary operation called “multiplication” (this means a map from G × G to G, where the image of (a;b) is usually written as the juxtaposition ab) satisfying (a) the associative law holds: a(bc) = (ab)c for all a;b;c 2 G; (b) there is an element e 2 G such that ea = ae = a for all a 2 G; (c) for any a 2 G, there is an element a∗ 2 G such that aa∗ = a∗a = e. The element e is unique, and for each a there is a unique a∗.(Exercise: Prove this!) We call e the identity, and usually write it as 1; and a∗ the inverse of a, usually written a−1. The associative law means that we can write a(bc) or (ab)c unambiguously as abc. It follows that the product of any number of elements is independent of the bracketing, so we can write a1a2 ···an unambiguously. (Exercise: Prove this!) Now any transformation group is an (abstract) group. (The essential part of the argument is that composition of transformations is always associative: writing maps on the right so that xa is the image of x 2 W under a 2 G, we have x(a(bc) = (xa)(bc) = ((xa)b)c) = (xab)c = x((ab)c): Conversely, given an abstract group G, there is a transformation group which is isomorphic to G, in the sense that the elements correspond in a bijective fash- ion and composition of transformations corresponds to multiplication of group elements. This famous result is Cayley’s theorem. I outline the proof. We take W = G and, for each a 2 G, let Ta be the operation of right multiplication by a: xTa = xa. Then • if a 6= b then 1Ta = a 6= b = 1Tb, so Ta 6= Tb; •f Ta : a 2 Gg is a transformation group (this is for you to prove); • xTaTb = (xa)Tb = (xa)b = x(ab) = xTab, so the transformations compose in the same way that the group elements multiply. 4 Chapter 1. Finite groups This theorem then closes the circle, and shows that groups as defined by the axioms are “the same thing” as transformation groups. Cayley’s name is also attached to another object, the Cayley table or multipli- cation table of a group. The two things are connected. Here is an example of a group of order 4: · g1 g2 g3 g4 g1 g1 g2 g3 g4 g2 g2 g1 g4 g3 g3 g3 g4 g1 g2 g4 g4 g3 g2 g1 This table gives the group multiplication: for example, to find the product of g2 and g3, we look in the row labelled g2 and the column labelled g3, and find the element g4. Now we see that the translations Ta just described above are given by the columns of this array. So, ignoring the letters g, the columns of the table give us a group of permutations isomorphic to the group we started with. These are 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ; ; ; 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 (the transformation corresponding to each symbol sends each element in the top row to that beneath it in the bottom row). In the more usual cycle notation (which we define later), they are (1)(2)(3)(4);(1;2)(3;4);(1;3)(2;4);(1;4)(2;3): Note that we use columns rather than rows because of our convention that maps are written on the right. (In the above case it doesn’t matter, since the group is commutative: ab = ba for all a;b 2 G (which is shown by the fact that the Cayley table, as a matrix, is symmetric). Why doesn’t an abstract group need a “closure” axiom? Multiplication is a binary operation, so by definition for any a;b 2 G there is a product ab 2 G. However, many authors give this as the first of four axioms for a group. We will see the reason for this later. Nowdays, “group” means “abstract group” in the sense defined above. The advantage is that it is independent of the means of representation.
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