Visual Group Theory PDF
Preview Visual Group Theory
i i \master" | 2011/10/18 | 17:10 | page i | #1 i i Visual Group Theory i i i i i i \master" | 2011/10/18 | 17:10 | page ii | #2 i i c 2009bytheMathematicalAssociationof America,Inc. (cid:13) LibraryofCongressCatalogCardNumber2009923532 Print edition ISBN978-0-88385-757-1 Electronic edition ISBN978-1-61444-102-1 PrintedintheUnited Statesof America Current Printing (last digit): 109876543 21 i i i i i i \master" | 2011/10/18 | 17:10 | page iii | #3 i i Visual Group Theory Nathan C. Carter Bentley University ® Published and Distributedby The MathematicalAssociation of America i i i i i i \master" | 2011/10/18 | 17:10 | page iv | #4 i i Council on Publications Paul Zorn, Chair Classroom Resource Materials EditorialBoard Gerald M. Bryce, Editor William C. Bauldry Diane L. Herrmann Loren D. Pitt Wayne Roberts Susan G. Staples Philip D. Straffin HollyS. Zullo i i i i i i \master" | 2011/10/18 | 17:10 | page v | #5 i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary classroom material forstudents|laboratoryexercises,projects,historicalinformation,textbookswithunusual approaches for presentingmathematical ideas, career information,etc. 101CareersinMathematics,2ndedition editedbyAndrew Sterrett Archimedes:WhatDid HeDoBesidesCryEureka?,ShermanStein CalculusMysteriesandThrillers,R. GrantWoods ConjectureandProof, Miklo(cid:19)sLaczkovich CreativeMathematics,H. S. Wall EnvironmentalMathematicsin theClassroom,edited byB.A. FusaroandP. C.Kenschaft ExploratoryExamplesforRealAnalysis,JoanneE. SnowandKirk E. Weller GeometryFromAfrica:MathematicalandEducationalExplorations,PaulusGerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz andKarenDee Michalowicz Identification NumbersandCheckDigit Schemes,JosephKirtland InterdisciplinaryLivelyApplicationProjects,edited byChrisArney InverseProblems:ActivitiesforUndergraduates,CharlesW. Groetsch LaboratoryExperiencesinGroupTheory,Ellen MaycockParker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz MathMadeVisual:CreatingImagesforUnderstandingMathematics,ClaudiAlsinaandRogerB. Nelsen OrdinaryDifferential Equations:ABriefEclecticTour,David A.Sa(cid:19)nchez OvalTrackandOtherPermutationPuzzles,JohnO. Kiltinen APrimerofAbstractMathematics,Robert B. Ash ProofsWithoutWords,Roger B. Nelsen ProofsWithoutWordsII, Roger B. Nelsen SheDoesMath!, editedbyMarla Parker SolveThis: MathActivitiesforStudentsandClubs,JamesS.Tanton StudentManualforMathematicsforBusinessDecisionsPart1:ProbabilityandSimulation, David Williamson, Marilou Mendel, JulieTarr, andDeborahYoklic StudentManualforMathematicsforBusinessDecisionsPart2:CalculusandOptimization,David Williamson, Marilou Mendel, JulieTarr, andDeborahYoklic TeachingStatisticsUsingBaseball,JimAlbert VisualGroupTheory,NathanC.Carter Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, GavinLaRose,ThomasRatliff, ElynRykken MAAService Center P.O. Box91112 Washington,DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i \master" | 2011/10/18 | 17:10 | page vi | #6 i i i i i i i i \master" | 2011/10/18 | 17:10 | page vii | #7 i i Acknowledgments I am grateful to God for life and breath and mathematics, as well as my ability to write it, draw it, and enjoy it. I am grateful to my family, especially Lydia, who put up with my absence during times I neededto work on this manuscript. Many thanks to Doug Hofstadter, who showed me the power of group theory visu- alization and supported my work on it in several ways, including much good advice. Many thanks to Charlie Hadlock, who kept me away from not-so-good advice (such as \Don’t write a book before you get tenure") and who guided many of the early steps of this project. I also thank Don Albers, the MAA, and Bentley University(particularly Rick Cleary and Kate Davy), all of whom encouraged, supported, and took a chance on a first-timeauthor. I must also thank those whose mathematical work helped me in mine. One of my early exposures to group theory visualization techniques was Magnus and Grossman’s GroupsandTheirGraphs.InwritingthisbookI madefrequentreferencetotheexcellent texts by Michael Artin, John Fraleigh, Charles Hadlock, and Thomas Hungerford. The powerful and well-designed software Asymptote by John Bowman helped me make the more than 300 figures herein. Thank you! Lastly, there are many peoplewho read early chapters and drafts and provided much helpfulfeedback. These include Doug Hofstadter, Jon Zivan, an anonymous referee, my parents, and my Fall 2006 Discrete Mathematics class, especially Kathryn Ogorzalek. Thank you for helping me see my work throughyour eyes, and improve it. vii i i i i i i \master" | 2011/10/18 | 17:10 | page viii | #8 i i i i i i i i \master" | 2011/10/18 | 17:10 | page ix | #9 i i Preface If you are interestedin learning about group theory in a relaxed, intuitiveway, then this bookisforyou.Isaylearningaboutgrouptheorybecausethisbookdoesnotaimtocover grouptheorycomprehensively.Hereinyouwillfindclear, illustratedexpositionaboutthe basics of the subject, which will give you a solid foundation of intuitions, images, and examples on which you can buildwith furtherstudy. Thisbookisidealforastudentbeginningafirstcourseingrouptheory.Itcanbeused inplace ofa traditionaltextbook, oras a supplementto one,butitsaim isquitedifferent thanthatofatraditionaltext.Mosttextbookspresentthetheoryofgroupsusingtheorems, proofs, and examples. Their exercises teach you how to make conjectures about groups and prove or refute them. This book, however, teaches you to know groups. You will see them,experimentwiththem,and understandtheirsignificance.Thementallibraryof images and intuitionsyou gain from reading this book will enable you to appreciate far betterthe facts and proofs in a traditionaltext. Thisbookis alsoappropriateforrecreationalreading.If youwantan overviewofthe theory of groups, or to learn key principles without going as deep as some upper-level undergraduatemathematicscourses, youcan read thisbookby itself.Onlya typicalhigh school mathematics education is assumed, but you should have a willingness to think analytically. My work on this book stems from Group Explorer, a software package I wrote that creates illustrations for finite groups, and allows the user to interact and experiment with them. Many of the illustrations in this text were generated with the help of Group Explorer, and the investigations possible in Group Explorer can help you with some of this book’s exercises. YoudonotneedGroupExplorer tobenefitfromthisbook;veryfewexercisesspecif- icallydirectyou toGroupExplorer. ButI recommendtakingfulladvantage ofhands-on, interactive learning experiences when they’re available; the more involved we are, the more we tendto learn. Group Explorer isfree software, available forall major operating systems. You can findit online at http://groupexplorer.sourceforge.net. ix i i i i
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