DOCSLIB.ORG

Appendix I - Group Theory I

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix I - Group Theory I 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' ..
Load more

Recommended publications
  • 18.703 Modern Algebra, Presentations and Groups of Small
    12. Presentations and Groups of small order Definition-Lemma 12.1. Let A be a set. A word in A is a string of 0 elements of A and their inverses. We say that the word w is obtained 0 from w by a reduction, if we can get from w to w by repeatedly applying the following rule, −1 −1 • replace aa (or a a) by the empty string. 0 Given any word w, the reduced word w associated to w is any 0 word obtained from w by reduction, such that w cannot be reduced any further. Given two words w1 and w2 of A, the concatenation of w1 and w2 is the word w = w1w2. The empty word is denoted e. The set of all reduced words is denoted FA. With product defined as the reduced concatenation, this set becomes a group, called the free group with generators A. It is interesting to look at examples. Suppose that A contains one element a. An element of FA = Fa is a reduced word, using only a and a−1 . The word w = aaaa−1 a−1 aaa is a string using a and a−1. Given any such word, we pass to the reduction w0 of w. This means cancelling as much as we can, and replacing strings of a’s by the corresponding power. Thus w = aaa −1 aaa = aaaa = a 4 = w0 ; where equality means up to reduction. Thus the free group on one generator is isomorphic to Z. The free group on two generators is much more complicated and it is not abelian.
    [Show full text]
  • Some Remarks on Semigroup Presentations
    SOME REMARKS ON SEMIGROUP PRESENTATIONS B. H. NEUMANN Dedicated to H. S. M. Coxeter 1. Let a semigroup A be given by generators ai, a2, . , ad and defining relations U\ — V\yu2 = v2, ... , ue = ve between these generators, the uu vt being words in the generators. We then have a presentation of A, and write A = sgp(ai, . , ad, «i = vu . , ue = ve). The same generators with the same relations can also be interpreted as the presentation of a group, for which we write A* = gpOi, . , ad\ «i = vu . , ue = ve). There is a unique homomorphism <£: A—* A* which maps each generator at G A on the same generator at G 4*. This is a monomorphism if, and only if, A is embeddable in a group; and, equally obviously, it is an epimorphism if, and only if, the image A<j> in A* is a group. This is the case in particular if A<j> is finite, as a finite subsemigroup of a group is itself a group. Thus we see that A(j> and A* are both finite (in which case they coincide) or both infinite (in which case they may or may not coincide). If A*, and thus also Acfr, is finite, A may be finite or infinite; but if A*, and thus also A<p, is infinite, then A must be infinite too. It follows in particular that A is infinite if e, the number of defining relations, is strictly less than d, the number of generators. 2. We now assume that d is chosen minimal, so that A cannot be generated by fewer than d elements.
    [Show full text]
  • Knapsack Problems in Groups
    MATHEMATICS OF COMPUTATION Volume 84, Number 292, March 2015, Pages 987–1016 S 0025-5718(2014)02880-9 Article electronically published on July 30, 2014 KNAPSACK PROBLEMS IN GROUPS ALEXEI MYASNIKOV, ANDREY NIKOLAEV, AND ALEXANDER USHAKOV Abstract. We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give var- ious examples of finitely presented groups where the subset sum problem is NP-complete. 1. Introduction 1.1. Motivation. This is the first in a series of papers on non-commutative discrete (combinatorial) optimization. In this series we propose to study complexity of the classical discrete optimization (DO) problems in their most general form — in non- commutative groups. For example, DO problems concerning integers (subset sum, knapsack problem, etc.) make perfect sense when the group of additive integers is replaced by an arbitrary (non-commutative) group G. The classical lattice problems are about subgroups (integer lattices) of the additive groups Zn or Qn, their non- commutative versions deal with arbitrary finitely generated subgroups of a group G. The travelling salesman problem or the Steiner tree problem make sense for arbitrary finite subsets of vertices in a given Cayley graph of a non-commutative infinite group (with the natural graph metric). The Post correspondence problem carries over in a straightforward fashion from a free monoid to an arbitrary group. This list of examples can be easily extended, but the point here is that many classical DO problems have natural and interesting non-commutative versions.
    [Show full text]
  • Combinatorial Group Theory
    Combinatorial Group Theory Charles F. Miller III March 5, 2002 Abstract These notes were prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2002 by C. F. Miller. Contents 1 Free groups and presentations 3 1.1 Free groups . 3 1.2 Presentations by generators and relations . 7 1.3 Dehn’s fundamental problems . 9 1.4 Homomorphisms . 10 1.5 Presentations and fundamental groups . 12 1.6 Tietze transformations . 14 1.7 Extraction principles . 15 2 Construction of new groups 17 2.1 Direct products . 17 2.2 Free products . 19 2.3 Free products with amalgamation . 21 2.4 HNN extensions . 24 3 Properties, embeddings and examples 27 3.1 Countable groups embed in 2-generator groups . 27 3.2 Non-finite presentability of subgroups . 29 3.3 Hopfian and residually finite groups . 31 4 Subgroup Theory 35 4.1 Subgroups of Free Groups . 35 4.1.1 The general case . 35 4.1.2 Finitely generated subgroups of free groups . 35 4.2 Subgroups of presented groups . 41 4.3 Subgroups of free products . 43 4.4 Groups acting on trees . 44 5 Decision Problems 45 5.1 The word and conjugacy problems . 45 5.2 Higman’s embedding theorem . 51 1 5.3 The isomorphism problem and recognizing properties . 52 2 Chapter 1 Free groups and presentations In introductory courses on abstract algebra one is likely to encounter the dihedral group D3 consisting of the rigid motions of an equilateral triangle onto itself.
    [Show full text]
  • Deforming Discrete Groups Into Lie Groups
    Deforming discrete groups into Lie groups March 18, 2019 2 General motivation The framework of the course is the following: we start with a semisimple Lie group G. We will come back to the precise definition. For the moment, one can think of the following examples: • The group SL(n; R) n × n matrices of determinent 1, n • The group Isom(H ) of isometries of the n-dimensional hyperbolic space. Such a group can be seen as the transformation group of certain homoge- neous spaces, meaning that G acts transitively on some manifold X. A pair (G; X) is what Klein defines as “a geometry” in its famous Erlangen program [Kle72]. For instance: n n • SL(n; R) acts transitively on the space P(R ) of lines inR , n n • The group Isom(H ) obviously acts on H by isometries, but also on n n−1 @1H ' S by conformal transformations. On the other side, we consider a group Γ, preferably of finite type (i.e. admitting a finite generating set). Γ may for instance be the fundamental group of a compact manifold (possibly with boundary). We will be interested in representations (i.e. homomorphisms) from Γ to G, and in particular those representations for which the intrinsic geometry of Γ and that of G interact well. Such representations may not exist. Indeed, there are groups of finite type for which every linear representation is trivial ! This groups won’t be of much interest for us here. We will often start with a group of which we know at least one “geometric” representation.
    [Show full text]
  • Combinatorial Group Theory
    Combinatorial Group Theory Charles F. Miller III 7 March, 2004 Abstract An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded many times for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2004 by C. F. Miller III. Contents 1 Preliminaries 3 1.1 About groups . 3 1.2 About fundamental groups and covering spaces . 5 2 Free groups and presentations 11 2.1 Free groups . 12 2.2 Presentations by generators and relations . 16 2.3 Dehn’s fundamental problems . 19 2.4 Homomorphisms . 20 2.5 Presentations and fundamental groups . 22 2.6 Tietze transformations . 24 2.7 Extraction principles . 27 3 Construction of new groups 30 3.1 Direct products . 30 3.2 Free products . 32 3.3 Free products with amalgamation . 36 3.4 HNN extensions . 43 3.5 HNN related to amalgams . 48 3.6 Semi-direct products and wreath products . 50 4 Properties, embeddings and examples 53 4.1 Countable groups embed in 2-generator groups . 53 4.2 Non-finite presentability of subgroups . 56 4.3 Hopfian and residually finite groups . 58 4.4 Local and poly properties . 61 4.5 Finitely presented coherent by cyclic groups . 63 1 5 Subgroup Theory 68 5.1 Subgroups of Free Groups . 68 5.1.1 The general case . 68 5.1.2 Finitely generated subgroups of free groups .
    [Show full text]
  • Chapter 1 GENERAL STRUCTURE and PROPERTIES
    Chapter 1 GENERAL STRUCTURE AND PROPERTIES 1.1 Introduction In this Chapter we would like to introduce the main de¯nitions and describe the main properties of groups, providing examples to illustrate them. The detailed discussion of representations is however demanded to later Chapters, and so is the treatment of Lie groups based on their relation with Lie algebras. We would also like to introduce several explicit groups, or classes of groups, which are often encountered in Physics (and not only). On the one hand, these \applications" should motivate the more abstract study of the general properties of groups; on the other hand, the knowledge of the more important and common explicit instances of groups is essential for developing an e®ective understanding of the subject beyond the purely formal level. 1.2 Some basic de¯nitions In this Section we give some essential de¯nitions, illustrating them with simple examples. 1.2.1 De¯nition of a group A group G is a set equipped with a binary operation , the group product, such that1 ¢ (i) the group product is associative, namely a; b; c G ; a (b c) = (a b) c ; (1.2.1) 8 2 ¢ ¢ ¢ ¢ (ii) there is in G an identity element e: e G such that a e = e a = a a G ; (1.2.2) 9 2 ¢ ¢ 8 2 (iii) each element a admits an inverse, which is usually denoted as a¡1: a G a¡1 G such that a a¡1 = a¡1 a = e : (1.2.3) 8 2 9 2 ¢ ¢ 1 Notice that the axioms (ii) and (iii) above are in fact redundant.
    [Show full text]
  • Preserving Torsion Orders When Embedding Into Groups Withsmall
    Preserving torsion orders when embedding into groups with ‘small’ finite presentations Maurice Chiodo, Michael E. Hill September 9, 2018 Abstract We give a complete survey of a construction by Boone and Collins [3] for em- bedding any finitely presented group into one with 8 generators and 26 relations. We show that this embedding preserves the set of orders of torsion elements, and in particular torsion-freeness. We combine this with a result of Belegradek [2] and Chiodo [6] to prove that there is an 8-generator 26-relator universal finitely pre- sented torsion-free group (one into which all finitely presented torsion-free groups embed). 1 Introduction One famous consequence of the Higman Embedding Theorem [11] is the fact that there is a universal finitely presented (f.p.) group; that is, a finitely presented group into which all finitely presented groups embed. Later work was done to give upper bounds for the number of generators and relations required. In [14] Valiev gave a construction for embedding any given f.p. group into a group with 14 generators and 42 relations. In [3] Boone and Collins improved this to 8 generators and 26 relations. Later in [15] Valiev further showed that 21 relations was possible. In particular we can apply any of these constructions to a universal f.p. group to get a new one with only a few generators and relations. More recently Belegradek in [2] and Chiodo in [6] independently showed that there exists a universal f.p. torsion-free group; that is, an f.p. torsion-free group into which all f.p.
    [Show full text]
  • The Compressed Word Problem for Groups
    Markus Lohrey The Compressed Word Problem for Groups – Monograph – April 3, 2014 Springer Dedicated to Lynda, Andrew, and Ewan. Preface The study of computational problems in combinatorial group theory goes back more than 100 years. In a seminal paper from 1911, Max Dehn posed three decision prob- lems [46]: The word problem (called Identitatsproblem¨ by Dehn), the conjugacy problem (called Transformationsproblem by Dehn), and the isomorphism problem. The first two problems assume a finitely generated group G (although Dehn in his paper requires a finitely presented group). For the word problem, the input con- sists of a finite sequence w of generators of G (also known as a finite word), and the goal is to check whether w represents the identity element of G. For the con- jugacy problem, the input consists of two finite words u and v over the generators and the question is whether the group elements represented by u and v are conju- gated. Finally, the isomorphism problem asks, whether two given finitely presented groups are isomorphic.1 Dehns motivation for studying these abstract group theo- retical problems came from topology. In a previous paper from 1910, Dehn studied the problem of deciding whether two knots are equivalent [45], and he realized that this problem is a special case of the isomorphism problem (whereas the question of whether a given knot can be unknotted is a special case of the word problem). In his paper from 1912 [47], Dehn gave an algorithm that solves the word problem for fundamental groups of orientable closed 2-dimensional manifolds.
    [Show full text]
  • Finitely Presented Groups 1
    Finitely presented groups 1 Max Neunhöffer LMS Short Course on Computational Group Theory 29 July – 2 August 2013 Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 1 / 12 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k.
    [Show full text]
  • Topics in Combinatorial Group Theory
    Topics in Combinatorial Group Theory Preface I gave a course on Combinatorial Group Theory at ETH, Zurich, in the Winter term of 1987/88. The notes of that course have been reproduced here, essentially without change. I have made no attempt to improve on those notes, nor have I made any real attempt to provide a complete list of references. I have, however, included some general references which should make it possible for the interested reader to obtain easy access to any one of the topics treated here. In notes of this kind, it may happen that an idea or a theorem that is due to someone other than the author, has inadvertently been improperly acknowledged, if at all. If indeed that is the case here, I trust that I will be forgiven in view of the informal nature of these notes. Acknowledgements I would like to thank R. Suter for taking notes of the course and for his many comments and corrections and M. Schunemann for a superb job of “TEX-ing” the manuscript. I would also like to acknowledge the help and insightful comments of Urs Stammbach. In addition, I would like to take this opportunity to express my thanks and appreciation to him and his wife Irene, for their friendship and their hospitality over many years, and to him, in particular, for all of the work that he has done on my behalf, making it possible for me to spend so many pleasurable months in Zurich. Combinatorial Group Theory iii CONTENTS Chapter I History 1. Introduction ..................................................................1 2.
    [Show full text]
  • Chapter I: Groups 1 Semigroups and Monoids
    Chapter I: Groups 1 Semigroups and Monoids 1.1 Definition Let S be a set. (a) A binary operation on S is a map b : S × S ! S. Usually, b(x; y) is abbreviated by xy, x · y, x ∗ y, x • y, x ◦ y, x + y, etc. (b) Let (x; y) 7! x ∗ y be a binary operation on S. (i) ∗ is called associative, if (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x; y; z 2 S. (ii) ∗ is called commutative, if x ∗ y = y ∗ x for all x; y 2 S. (iii) An element e 2 S is called a left (resp. right) identity, if e ∗ x = x (resp. x ∗ e = x) for all x 2 S. It is called an identity element if it is a left and right identity. (c) The set S together with a binary operation ∗ is called a semigroup if ∗ is associative. A semigroup (S; ∗) is called a monoid if it has an identity element. 1.2 Examples (a) Addition (resp. multiplication) on N0 = f0; 1; 2;:::g is a binary operation which is associative and commutative. The element 0 (resp. 1) is an identity element. Hence (N0; +) and (N0; ·) are commutative monoids. N := f1; 2;:::g together with addition is a commutative semigroup, but not a monoid. (N; ·) is a commutative monoid. (b) Let X be a set and denote by P(X) the set of its subsets (its power set). Then, (P(X); [) and (P(X); \) are commutative monoids with respective identities ; and X. (c) x∗y := (x+y)=2 defines a binary operation on Q which is commutative but not associative.
    [Show full text]