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DISCOVERING GROUP THEORY A Transition to Advanced Mathematics www.TechnicalBooksPDF.com TEXTBOOKS in MATHEMATICS Series Editors: Al Boggess and Ken Rosen PUBLISHED TITLES ABSTRACT ALGEBRA: A GENTLE INTRODUCTION Gary L Mullen and James A Sellers ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION William Paulsen ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH Jonathan K Hodge, Steven Schlicker, and Ted Sundstrom ADVANCED LINEAR ALGEBRA Hugo Woerdeman ADVANCED LINEAR ALGEBRA Nicholas Loehr ADVANCED LINEAR ALGEBRA, SECOND EDITION Bruce Cooperstein APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION Richard Klima, Neil Sigmon, and Ernest Stitzinger APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE Vladimir Dobrushkin A BRIDGE TO HIGHER MATHEMATICS Valentin Deaconu and Donald C Pfaff COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH MATLAB® AND MPI, SECOND EDITION Robert E White A COURSE IN DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS, SECOND EDITION Stephen A Wirkus, Randall J Swift, and Ryan Szypowski A COURSE IN ORDINARY DIFFERENTIAL EQUATIONS, SECOND EDITION Stephen A Wirkus and Randall J Swift DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION Steven G Krantz www.TechnicalBooksPDF.com PUBLISHED TITLES CONTINUED DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH BOUNDARY VALUE PROBLEMS Steven G Krantz DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES, THIRD EDITION George F Simmons DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY Mark A McKibben and Micah D Webster ELEMENTARY NUMBER THEORY James S Kraft and Lawrence C Washington EXPLORING CALCULUS: LABS AND PROJECTS WITH MATHEMATICA® Crista Arangala and Karen A Yokley EXPLORING GEOMETRY, SECOND EDITION Michael Hvidsten EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA® Crista Arangala EXPLORING THE INFINITE: AN INTRODUCTION TO PROOF AND ANALYSIS Jennifer Brooks GRAPHS & DIGRAPHS, SIXTH EDITION Gary Chartrand, Linda Lesniak, and Ping Zhang INTRODUCTION TO ABSTRACT ALGEBRA, SECOND EDITION Jonathan D H Smith INTRODUCTION TO MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS, SECOND EDITION Charles E Roberts, Jr INTRODUCTION TO NUMBER THEORY, SECOND EDITION Marty Erickson, Anthony Vazzana, and David Garth LINEAR ALGEBRA, GEOMETRY AND TRANSFORMATION Bruce Solomon MATHEMATICAL MODELLING WITH CASE STUDIES: USING MAPLE™ AND MATLAB®, THIRD EDITION B Barnes and G R Fulford MATHEMATICS IN GAMES, SPORTS, AND GAMBLING–THE GAMES PEOPLE PLAY, SECOND EDITION Ronald J Gould THE MATHEMATICS OF GAMES: AN INTRODUCTION TO PROBABILITY David G Taylor www.TechnicalBooksPDF.com PUBLISHED TITLES CONTINUED A MATLAB® COMPANION TO COMPLEX VARIABLES A David Wunsch MEASURE AND INTEGRAL: AN INTRODUCTION TO REAL ANALYSIS, SECOND EDITION Richard L Wheeden MEASURE THEORY AND FINE PROPERTIES OF FUNCTIONS, REVISED EDITION Lawrence C Evans and Ronald F Gariepy NUMERICAL ANALYSIS FOR ENGINEERS: METHODS AND APPLICATIONS, SECOND EDITION Bilal Ayyub and Richard H McCuen ORDINARY DIFFERENTIAL EQUATIONS: AN INTRODUCTION TO THE FUNDAMENTALS Kenneth B Howell PRINCIPLES OF FOURIER ANALYSIS, SECOND EDITION Kenneth B Howell REAL ANALYSIS AND FOUNDATIONS, FOURTH EDITION Steven G Krantz RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION Bilal M Ayyub SPORTS MATH: AN INTRODUCTORY COURSE IN THE MATHEMATICS OF SPORTS SCIENCE AND SPORTS ANALYTICS Roland B Minton TRANSFORMATIONAL PLANE GEOMETRY Ronald N Umble and Zhigang Han www.TechnicalBooksPDF.com TEXTBOOKS in MATHEMATICS DISCOVERING GROUP THEORY A Transition to Advanced Mathematics Tony Barnard Hugh Neill www.TechnicalBooksPDF.com CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20160725 International Standard Book Number-13: 978-1-138-03016-9 (Paperback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging‑in‑Publication Data Names: Barnard, Tony (Mathematics professor) | Neill, Hugh | Barnard, Tony (Mathematics professor) Mathematical groups Title: Discovering group theory / Tony Barnard and Hugh Neill Other titles: Mathematical groups Description: Boca Raton : CRC Press, 2017 | Previous edition: Mathematical groups / Tony Barnard and Hugh Neill (London : Teach Yourself Books, 1996) | Includes index Identifiers: LCCN 2016029694 | ISBN 9781138030169 Subjects: LCSH: Group theory Textbooks | Algebra Textbooks | Mathematics Study and teaching Classification: LCC QA174.2 B37 2017 | DDC 512/.2 dc23 LC record available at https://lccn.loc.gov/2016029694 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com www.TechnicalBooksPDF.com Contents Preface .xi Proof 1.1 The Need for Proof 1.2 Proving by Contradiction 1.3 If, and Only If .4 1.4 Definitions .6 1.5 Proving That Something Is False .6 1.6 Conclusion .7 What You Should Know Exercise Sets 2.1 What Is a Set? 2.2 Examples of Sets: Notation 2.3 Describing a Set 10 2.4 Subsets 11 2.5 Venn Diagrams 12 2.6 Intersection and Union 13 2.7 Proving That Two Sets Are Equal 14 What You Should Know 16 Exercise 16 Binary Operations 19 3.1 Introduction 19 3.2 Binary Operations 19 3.3 Examples of Binary Operations 20 3.4 Tables 21 3.5 Testing for Binary Operations 22 What You Should Know 23 Exercise 23 Integers 25 4.1 Introduction 25 4.2 The Division Algorithm 25 4.3 Relatively Prime Pairs of Numbers 26 4.4 Prime Numbers 27 4.5 Residue Classes of Integers 28 4.6 Some Remarks 32 What You Should Know 32 Exercise 33 vii www.TechnicalBooksPDF.com viii Contents Groups 35 5.1 Introduction 35 5.2 Two Examples of Groups 35 5.3 Definition of a Group 37 5.4 A Diversion on Notation 39 5.5 Some Examples of Groups 40 5.6 Some Useful Properties of Groups 43 5.7 The Powers of an Element 44 5.8 The Order of an Element 46 What You Should Know 49 Exercise 49 Subgroups 51 6.1 Subgroups 51 6.2 Examples of Subgroups 52 6.3 Testing for a Subgroup 53 6.4 The Subgroup Generated by an Element 54 What You Should Know 56 Exercise 56 Cyclic Groups 59 7.1 Introduction 59 7.2 Cyclic Groups 60 7.3 Some Definitions and Theorems about Cyclic Groups 61 What You Should Know 63 Exercise 63 Products of Groups 65 8.1 Introduction 65 8.2 The Cartesian Product .65 8.3 Direct Product Groups 66 What You Should Know 67 Exercise 67 Functions 69 9.1 Introduction 69 9.2 Functions: A Discussion 69 9.3 Functions: Formalizing the Discussion 70 9.4 Notation and Language 71 9.5 Examples 71 9.6 Injections and Surjections 72 9.7 Injections and Surjections of Finite Sets 75 What You Should Know 77 Exercise 77 www.TechnicalBooksPDF.com ix Contents 10 Composition of Functions 81 10.1 Introduction 81 10.2 Composite Functions 81 10.3 Some Properties of Composite Functions 82 10.4 Inverse Functions .83 10.5 Associativity of Functions 86 10.6 Inverse of a Composite Function 86 10.7 The Bijections from a Set to Itself 88 What You Should Know 89 Exercise 10 89 11 Isomorphisms 91 11.1 Introduction 91 11.2 Isomorphism 93 11.3 Proving Two Groups Are Isomorphic 95 11.4 Proving Two Groups Are Not Isomorphic 96 11.5 Finite Abelian Groups 97 What You Should Know 102 Exercise 11 102 12 Permutations 105 12.1 Introduction 105 12.2 Another Look at Permutations 107 12.3 Practice at Working with Permutations 108 12.4 Even and Odd Permutations 113 12.5 Cycles 118 12.6 Transpositions 121 12.7 The Alternating Group 123 What You Should Know 124 Exercise 12 125 13 Dihedral Groups 127 13.1 Introduction 127 13.2 Towards a General Notation 129 13.3 The General Dihedral Group Dn 131 13.4 Subgroups of Dihedral Groups 132 What You Should Know 134 Exercise 13 134 14 Cosets 137 14.1 Introduction 137 14.2 Cosets 137 14.3 Lagrange’s Theorem 140 14.4 Deductions from Lagrange’s Theorem 141 www.TechnicalBooksPDF.com 205 Answers Chapter 12 1 ab =  3 4 5 , 5 1 ba =  1 5 , 3 1 a 2b =  4 5 , 2 1 ac −1 =  2 3 5 , 5 1 ( ac)−1 =  4 3 5 , 5 1 c −1ac =  3 4 5 1 1 x = a −1b =  5 2 3 5 , 4 5 3 1 x = a −1cb −1 =  4 8, 4, 3, 5, and Using Theorem 40, even, odd, odd Odd, odd (135)(24), (1342), (14)(25) 1 a  2 3 5 6 4 3 4 5 6 1 1 c  4 5 6 3 1 b  206 Answers a (1473) b (1465732) c (14)(23) d (123456) Align the rectangle in the same way as in the answer to question of Exercises Label the vertices with integer in the top righthand corner, and proceed clockwise with integers 2, 3, and Let f be the function from the group of symmetries of the rectangle to S4 defined by  fφ =   φ (1) φ ( 2) φ (3)  φ ( ) Then, with the notation of the answer to question in Exercises 5, f(I) = e, f(X) = (12)(34), f(Y) = (14)(23), and f(R) = (13)(24) e (12)(34)) (14)(23) (13)(24) e e (12)(34) (14)(23) (13)(24) (12)(34) (12)(34) e (13)(24) (14)(23) (14)(23) (14)(23) (13)(24) e (12)(34) (13)(24) (13)(24) (14)(23) (12)(34) e 10 Z 4 × S3 has two elements of order 3, namely, (0, (123)) and (0, (132)) Moreover, these are the only such elements, for, using the result of Exercise 8, question 10, if (x, y) ∈ Z 4 × S3 has order 3, then the order of x in Z is or 3, and hence 1, and the order of y ∈ S3 is or 3, and hence On the other hand, S4 has eight elements of order 3, (123), (132), (124), (142), (134), (143), (234), and (243) Similarly, Z2 × Z2 × S3 has only two elements of order 3, (0, 0, (123)) and (0, 0, (132)) Therefore, using the result of Exercise 11, question 5, S4 ≇ Z 4 × S3 and S4 ≇ Z2 × Z2 × S3 11 Let x = (a1a2…an) be of length n Then x2 = (a1a3…), x3 = (a1a4 …), …, xn−1 = (a1an…) are all different, and xn = e Therefore, n is the smallest power of x which gives the identity, so the order of x is n 12 Let a = anan−1 … a1 be a permutation made up of disjoint cycles of lengths α1, α 2, …, αn, and let l = LCM(α1, α 2, …, αn) Then al = e because all the disjoint cycles commute, and the order of each of them divides l Suppose that ar = e Then, since disjoint cycles commute, e = a r = an r an −1r … a1r Suppose that air ≠ e for some i ∈ {1, 2, …, n} Then there is a symbol, x say, such that air ( x) ≠ x But since the cycles are disjoint, no other cycle affects x, so ar(x) ≠ x This contradicts the fact that ar = e Therefore air = e for all i = 1, 2, …, n Therefore, r is a Answers 207 multiple of each of α1, α 2, …, αn Therefore l ≤ r, and so the order of a is l 13 From the result at the beginning of Section 12.6, every permutation x ∈ Sn can be written as a product of transpositions But any transposition (ab) can be written as a product of the form (ab) = (1a)(1b)(1a) Combining these two results shows that every permutation x ∈ Sn can be written as a product of the transpositions (12), (13), …, (1n) 14 Let a = (12…n − 1) and b = (n − 1  n) By direct calculation you can see that aba−1 = (1n), a2ba−2 = (2n), and, in general, proof by induction, aiba−i = (in) You can now use the result of question 13 15 Well defined The function f is well defined, because, since x, by virtue of being a member of An, is an even permutation and (12) is an odd permutation; their product, by Theorem 40, is an odd permutation, and therefore a member of Sn − An The mapping is therefore well defined Injection Suppose that f(x) = f(y) Then (12)x = (12)y Therefore, multiplying by (12), x = y, so f is an injection Surjection Let y ∈ Sn − An Then y is odd and so (12)y is even As f((12)y) = (12)((12)y) = ((12)(12))y = y, f is a surjection Thus f is a bijection, so An and Sn − An have the same number of elements But An and Sn − An are disjoint, and their union is Sn, which has n! elements So the number of elements in An is 12 n! Chapter 13 You can carry out a verification using a diagram (ba)2 = (ba)(ba) = b(aba) = bb = e Also (bai )2 = (bai )(bai ) = b(aibai ) = b(ai−1(aba)ai−1) = b(ai−1bai−1) = … = bb = e a a(ba) = aba = b b a−1 = an−1 c From question 1, since (ba)2 = e, (ba)−1 = ba d bab−1a−1 = b(aba)a−1a−1 = bba−2 = an−2 e (ba)(ba2) = b(aba)a = bba = a The proof is by induction Note that the basis step is true for i = 1 because ab = ab(aa−1) = (aba)a−1 = ba−1 = ban−1 i n−i Suppose that a b = ba is true for i = k Then akb = ban−k and k +1 k n− k a b = a( a b) = a(ba ) = ( ab)a n − k = (ba −1 )a n − k = ba n −(k + 1) So, if the statement is true for k, it is also true for k + 1 Therefore, by the principle of mathematical induction, the statement is true for all i ≥ 1, and so for i ∈ {1, 2, 3, …, n − 1} 208 Answers The answer depends on whether n is even or odd When n is odd, there are n subgroups of order 2, all of the form {e, bai} for i = {0, 1, …, n − 1} If n is even, then there is also the subgroup {e, an/2} If bai = baj, then, by the cancellation rule, Theorem 14, part 4, ai = aj and therefore e = aj−i, so j = i If bai = aj, then b = aj−i, so b is a power of a This is a contradiction, so bai ≠ aj for any i, j There are eight elements which have order These are the rotations X, Y, Z, T, X2, Y 2, Z2, and T 2 Note that X3 = Y3 = Z3 = T 3 = I The three half-turns, A, B, and C, about the mid-points of opposite edges, are of order so A2 = B2 = C2 = I This gives the 12 rotational symmetries I, A, B, C, X, Y, Z, T, X2, Y2, Z2, and T2 The proper subgroups of G are {I, A}, {I, B}, {I, C}, {I, A, B, C}, {I, X, X2}, {I, Y, Y 2}, {I, Z, Z2}, and {I, T, T 2} Suppose that a subgroup contains I, A, and X Then it also contains X2, AX = Z, XA = T, and hence Z2 and T There are already more than six elements In fact, it turns out that this subgroup must be the whole group This is true for any other starting set of the type I, A, and X   Labeling the vertices 1, 2, 3, and as in Figure 13.7, let f: G → S4 be defined by  f (φ ) =   φ (1) φ ( 2) φ (3)  φ ( )   Then f(I) = e, f(A) = (12)(34), f(B) = (13)(24), f(C) = (14)(23), f(X) = (234), f(Y) = (143), f(Z) = (124), f(T) = (132), f(X2) = (243), f(Y2) = (134), f(Z2) = (142), and f(T 2) = (123) All these permutations are even As there are 12 of them, and the order of A4 is 12, they constitute the whole of A4 and, as in Example 12.3.4, G ≅ A4 Let B = A ∩ Cn(= {e, as, a2s, a(d−1)s}) Then you need to show that A = B ∪ bamB   It is clear that B ∪ bamB ⊆ A since B ⊆ A and bam ∈ A To prove that A ⊆ B ∪ bamB, let x ∈ A If x ∈ B, then x ∈ B ∪ bamB, so suppose that x ∉ B Then x ∉ Cn Then bx ∈ Cn Therefore a−mbx ∈ Cn, because am ∈ Cn But a−mb = (bam)−1 ∈ A Therefore, a−mbx ∈ A, because x ∈ A Therefore a−mbx ∈ A ∩ Cn = B Therefore x = (bam)(a−mbx) ∈ bamB Chapter 14 The left cosets of {e, a2} are {e, a2}, {a, a3}, {b, ba2}, and {ba, ba3} The right cosets are {e, a2}, {a, a3}, {b, ba2}, and {ba, ba3} The left cosets of {e, b} are {e, b}, {a2, ba2}, {a, ba3}, and {a3, ba} The right cosets are {e, b}, {a2, ba2}, 209 Answers {a, ba}, and {a3, ba3} In the first case, the left cosets and the right cosets are identical; in the second case they are not First observe that the order of Z2 × Z3 is 6, and the order of Z2 × {0} is 2, so that there are three cosets The elements of Z2 × {0} are (0, 0) and (1, 0) The elements of Z2 × Z3 are (0, 0), (1, 0), (0, 1), (1, 1), (0, 2), and (1, 2)   The cosets of Z2 × {0} are (0, 0) + Z2 × {0} = Z2 × {0}, (1, 0) + Z2 × {0} = Z2 × {0}, (0, 1) + Z2 × {0} = Z2 × {1}, (1, 1) + Z2 × {0} = Z2 × {1}, (0, 2) + Z2 × {0} = Z2 × {2}, (1, 2) + Z2 × {0} = Z2 × {2}   Here are the calculations for the last coset written out in full (1, 2) + Z2 × {0} = (1, 2) + {(0, 0), (1, 0)} = {(1, 2), (0, 2)} = Z2 × {2} The left cosets of 4Z are 4Z, 1 + 4Z, 2 + 4Z, and 3 + 4Z Define f: {left cosets} → {right cosets} by f(xH) = (Hx −1) for all x ∈ G You need to show that f is well defined That is, you need to show that if xH and yH are the same coset, then they map to the same image That is Hx−1 = Hy−1 Well defined If xH = yH, then, by Theorem 45, part 1, x −1y ∈ H But you could prove a similar theorem for right cosets, that is, a necessary and sufficient condition for the right cosets Hx and Hy to be equal is xy −1 ∈ H Therefore, using this theorem, a necessary and sufficient condition for the cosets Hx −1 and Hy −1 to be equal is −1 −1 x −1 y −1 ∈ H But x −1 y −1 = x −1 y , so the conditions are identical Thus, if xH = yH, then Hx −1 = Hy −1 So f is well defined  Injection If f(xH) = f(yH) then Hx−1 = Hy−1 Therefore ( x −1 )( y −1 )−1 ∈ H as above, so x−1y ∈ H But then, by Theorem 45 part 1, it follows that xH = yH Therefore f is an injection  Surjection Let Hx be a right coset of H Then f ( x −1H ) = H ( x −1 )−1 = Hx , so f is surjective Therefore f is a bijection The cosets are of the form x + Z, where 0 ≤ x 

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