VisualGroupTheory

群论的可视化，英文版：If you are interested in learning about group theory in a relaxed, intuitive way, then this book is for you.master”-2011/1018-17:10— page 1-#2c 2009 by the Mathematical association of america, IncLibrary of Congress Catalog Card Number 2009923532Print edition isbn 978-0-88385-757-1Electronic edition isbn 978-1-61444-102-1Printed in the United states of mericaCurrent Printing (last digit)10987654321master”-2011/10/1817:10— page lll#Visual Group TheoryNathan C. CarterBentley universityPublished and distributed byhe Mathematical association of americamaster”-2011/10/18—17:10— page IV-#4Council on publicationsPaul Zorn ChairClassroom Resource materials Editorial boardGerald m. bryce editorWilliam c. bauldrvDiane l. herrmannLoren d. pittWayne robertspPhilip d. StraffiHolly S. zullomaster20110/18-17:10— page v-#5CLASSROOM RESOURCE MATERIALSClassroom Resource Materials is intended to provide supplementary classroom materialfor students--laboratory exercises, projects, historical information, textbooks with unusualapproaches for presenting mathematical ideas, career information, etc101 Careers in Mathematics, 2nd edition edited by Andrew SterrettArchimedes. What Did He Do Besides Cry Eureka? Sherman Steinand Thrillers. R. Grant WoodConjecture and Proof, Miklos LaczkovichCreative Mathematics. H.S. WallEnvironmental Mathematics in the Classroom edited by b. A. Fusaro and P. C. KenschaftExploratory Examples for Real Analysis, Joanne E. Snow and Kirk e. WellerGeometry From Africa. Mathematical and Educational Explorations, Paulus gerdesHistorical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katzand Karen dee michalowiczIdentification Numbers and Check Digit Schemes, Joseph KirtlandLively Application Projects, edited by chris aInverse Problems: Activities for Undergraduates, Charles W. GroelschLaboratory Experiences in Group Theory, Ellen Maycock ParkerLearn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and VictoratzMath Made visual: Creating Images for Understanding Mathematics, Claudi Alsina and roger bNelsenOrdinary Differential Equations: A Brief Eclectic Tour, David A S anchezOval track and Other permutation puzzles john o, KiltinenA Primer of abstract Mathematics, robert b. ashProofs Without Words, Roger B. NelsenProofs Without Words lL, Roger B. NelsenShe Does math!, edited hy Marla ParkerSolve This. Math Activities for Students and Clubs, James S. TantonStudent Manual for Mathematics for Business Decisions Part 1: Probability and Simulation, DavidWilliamson Marilou mendel. Julie Tarr. and Deborah YoklicStudent Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, DavidWilliamson Marilou mendel. Julie tarr. and Deborah YoklicTeaching Statistics Using Baseball, Jim AlbertVisual Group Theory, Nathan C. CarterWriting Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, AnnaliCrannell, Gavin LaRose, Thomas Ratliff, Elyn RykkenMAA Service CenterP O. Box 91112Washington. DC 20090-11121-800-331-1 MAA FAX:1-3012069789master”-2011/10/18-17:10— page vI—#6master2011/1018—17:10— page vll一#7AcknowledgmentsI am grateful to God for life and breath and mathematics, as well as my ability to writeit, draw it, and enjoy it. I am grateful to my family, especially Lydia, who put up withmy absence during times I needed to work on this manuscriptMany 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 adviceMany thanks to Charlie Hadlock, who kept me away from not-so-good advice(such asDon't write a book before you get tenure?) and who guided many of the early stepsof this project. I also thank Don Albers, the MAa, and Bentley University(particularlyRick Cleary and Kate Davy), all of whom encouraged, supported, and took a chance ona first-time authorI must also thank those whose mathematical work helped me in mine. One of myearly exposures to group theory visualization techniques was Magnus and Grossman'sGroups and Their Graphs. In writing this book I made frequent reference to the excellenttexts by Michael Artin, John Fraleigh, Charles Hadlock, and Thomas Hungerford. Thepowerful and well-designed software Asymptote by John Bowman helped me make themore than 300 figures herein. Thank you!Lastly, there are many people who read early chapters and drafts and provided muchhelpful feedback. These include Doug Hofstadter, Jon Zivan, an anonymous referee, myparents, and my Fall 2006 Discrete Mathematics class, especially Kathryn OgorzalekThank you for helping me see my work through your eyes, and improve itmaster2011/10/1817:10— page vi#8master”-2011/10/18-17:10- page Ix-#9PrefaceIf you are interested in learning about group theory in a relaxed, intuitive way, then thisbook is for you. I say learning about group theory because this book does not aim to covergroup theory comprehensively. Herein you will find clear, illustrated exposition about thebasics of the subject, which will give you a solid foundation of intuitions, images, andexamples on which you can build with further studyThis book is ideal for a student beginning a first course in group theory It can be usedin place of a traditional textbook, or as a supplement to one, but its aim is quite differentthan that of a traditional text. Most textbooks present the theory of groups using theoremsproofs, and examples. Their exercises teach you how to make conjectures about groupsand prove or refute them. This book, however, teaches you to know groups. You willsee them, experiment with them and understand their significance. The mental library ofimages and intuitions you gain from reading this book will enable you to appreciate farbetter the facts and proofs in a traditional textThis book is also appropriate for recreational reading. If you want an overview of thetheory of groups, or to learn key principles without going as deep as some upper-levelundergraduate mathematics courses, you can read this book by itself. Only a typical highschool mathematics education is assumed, but you should have a willingness to thinkMy work on this book stems from Group Explorer, a software package I wrote thatcreates illustrations for finite groups, and allows the user to interact and experimentwith them. Many of the illustrations in this text were generated with the help of GroupExplorer, and the investigations possible in Group Explorer can help you with some ofthis book’ s exercisesYou do not need Group Explorer to benefit from this book; very few exercises specif-ically direct you to Group Explorer. But I recommend taking full advantage of hands-on,interactive learning experiences when theyre available; the more involved we are, theInore we tend to learn. Group Explorer is free software, available for all major operatingsystems. You can find it online athttp://groupexplorer.sourceforge.et