Appendix I - Group theory I. EqUivalence relation 2. Groups and a bit of ring theory 3. Free group 4. Presentations of groups 5. Word problem 6. Reidemeister-Schreier method, presentation of a subgroup 7. Triangle groups In order to build and enhance mathematical theories, it is usually necessary to borrow and apply concepts from various branches of mathematics. This book is not different in this regard. However, it would be rather cumbersome, even nowadays electronically, for the reader to keep on popping off to the library to find an explanation of some bit of group theory or topology. So, in the first three appendices, we have gathered together the concepts and ideas that should make this book, on the whole, self-contained. In this appendix, besides providing a quick introduction to the fundamental concepts of group theory, we also cover, in some detail, less familiar group the­ ory, namely, the free group, the presentation of a group, and the Reidemeister­ Schreier method. Even though some of these concepts might be known to the reader, hopefully, the setting and their use in the theory of braids might shine a new perspective on these concepts . • Equivalence relation "Equivalence?" Throughout this book we say braids are equivalent, equal and even on occasion the same. To understand completely the mathematical rigour of equivalence, we need to precisely comprehend, how one object relates to another. Hence, we need to introduce the fundamental concept of an equivalence relation. Let us begin rather abstractly, suppose "E is a non-empty set and there exists among the elements of "E some way of relating one to another. EXAMPLE 1.1 (1) Let "E be the set of all real numbers, namely "E = lR.. We say that an element a of "E is related to another element b, if a is larger than b, or symbolically a > b. (2) Let us suppose "E is the set of all integers, so "E = ~. On this occasion, let us choose an arbitrary non-zero integer, n say. We can introduce a relation on "E as follows, an integer a is related to another integer b if a - b is divisible by n. In such circumstances, it is usual to say a is congruent to b modulo n, or in terms of mathematical notation a == b (mod n). Clearly, the above relations are not unique for their respective sets "E. Again, let us be a bit more abstract and define the notation a rv b if a is related to b 219 220 ApPENDIX I - GROUP THEORY in E. In the first example, a --- b if a > b, while in the second example, a --- b if a - b = nk for some integer k. DEFINITION 1.1 A relation on a set E is said to be an equivalence relation if the Ifollowing 3 conditions are satisfied: (1) (Reflective) a is related to itself, Le., a --- a. (2) (Symmetrie) If a --- b then b --- a. (3) (Transitive) If a --- band b --- c then a --- c. ExAMPLE 1.1 (1) (continued) Clearly, a > a cannot hold. Therefore, the relation given by a > b cannot be an equivalence relation. (2) (continued) We leave it as a short exercise for the reader to show that congru­ ence is an equivalence relation. The notion of an equivalence relation may be thought of as the generalization of the mathematieal relation "is equal to." Hence, it should be understandable, why we may blur on occasion the distinction between equal and equivalence. Now, suppose we are given a set E with an equivalence relation. If we fix an element a in E (a E E), then we can create a subset of E by just taking all the elements that are equivalent to a. Such a subset is called an equivalence class represented by a and denoted by [al. In terms of the notation, [al consists of all the elements xE E such that a rv X. EXAMPLE 1.1 (2) (continued) Let us fix n = 5. Then, [0] is the set {O, ±5, ±1O, ... }, [3] is the set { ... ,-7, -2,3,8 ... }. PRoPOSITION 1.1 Suppose E is a set with an equivalence relation. If [al n [bI =10, then [al = [bI. The significance of the above proposition is that we may completely decompose E into a collection of (disjoint ) equivalence classes. We shall denote this collection by E/rv • EXAMPLE 1.1 (2) (continued) Again, fix n = 5. Then E/rv has only 5 elements, E/rv = {[O], [1], [2], [3], [4]}. (1.1) Note that, [0] = [5] = [10] = ... , et cetera. EXERCISE 1.1 Let E be a set and suppose we can decompose E into disjoint subsets Al, A2 , ••• (the number of such subsets need not be finite). For any x, y E E, let us define a relation in E as follows, x rv y if x and y both belong to the same set, Ai say. Show that the above relation is an equivalence relation and (1.2) §2. GROUPS AND A BIT OF RING THEORY 221 As a final note to this section, suppose we have an equivalence relation on a set I:, then if a is related to b we have by definition that b is related to a. Hence, we may say without ambiguity that a and bare related. 11 Groups and a bit of ring theory IDEFINITION 2.1 A non-empty (finite or infinite) set, G, is said to be a group if the set G satisfies the following conditions: (1) To each (ordered) pair (a, b), consisting of elements a and b in G, we can assign a unique element c in G. The assignment of c to an ordered pair (a, b) is usually called a binary operation. For simplicity, c is usually called the product of a and b, and denoted by a 0 b or just by ab. (2) The product is associative, symbolically, (a 0 b) 0 c = a 0 (b 0 c). (3) There exists an element e in G such that for anyelement xE G, xoe = x and e 0 x = x. Such an element e is called the identity (element) of G. We shall usually denote e by 1. ( 4) For each element x in G, there is an element y in G such that x 0 y = e and y 0 x = e. Under these circumstances, y is said to be the inverse of x and denoted by X-I. EXERCISE 2.1 (1) Show that even though it was not explicitly stated the identity, e, and the inverse element are both unique. (2) Show that, for x and y in a group G, (xy)-1 = y-lx -l and (x-1)-1 = X. Even a cursory glance immediately indicates that the essence of a group is the binary operationjproduct. However, a set G need not have just one way of assigning a product. So, a set with two possible products gives rise to 2 groups. If such a case arises, we will explicitly delineate the 2 products. For the most part, if the product is readily understood, we will not differentiate in our notation between the set G and the group G. EXAMPLE 2.1 (1) The set of all integers, 2, forms a group with the binary opera­ tion given by the ordinary addition. Similarly, the rational numbers, Q, the real numbers, lR, and the complex numbers, (C, are also groups. (2) The set of all non-zero rational numbers, QX, non-zero real numbers, lR x , and non-zero complex numbers, (Cx, is each a group with binary operationjproduct given by the usual multiplication. However, the set of non-zero integers does not form a group with multiplication as the product, for it is impossible to find an inverse for any integer besides l. EXAMPLE 2.2 Let G be the set of all permutations on n numbers 1,2, ... ,n, with n ~ 2. It is standard to denote an element, (7, of G as (2.1) where the notation indicates that j permutes to the integer i j . 222 ApPENDIX I - GROUP THEORY Suppose T is another element of G, Le., (2.2) Clearly, we may rearrange the columns of T so that (2.3) Hence, we may define a product on G as ~) . (2.4) Caveat leetor, The above product is performed from left to right. However, there is nothing in our definition of a binary operation that precludes us from starting with T and then rearranging 0'. In this case, the product would be from right to left. Such a distinction can at times be very important. The set G with the above product forms a group, in which the identity element is just the identity permutation, obtained by setting ij = j in (2.1), while the inverse of 0', denoted by 0'-1, is (2.5) This group G is more commonly known as the symmetrie group on n letters and denoted by Sn. While the notation above for apermutation is explicit in its clarity, at times we will employ a shorthand version of the above notation. Consider the row of r integers (al a2 ... ar ), this may be thought of a representing apermutation with ai permutating to ai+l for i = 1,2, ... ,r -1 and ar permutes to al' Such a row representation of apermutation is usually called a eycle. The product of cycles is just a matter of following, moving from left to right, the permutation of ai, say, along each consecutive cycle, Le., in the first cycle ai permutes to ai+l, in the second cycle ai+l permutes to ah' in the third cycle ah permutes to ajs' ..