An Introduction to Group Theory
Applications to Mathematical Music Theory
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About the book
Description
The success of Group Theory is impressive and extraordinary. Its influence is strongly felt in almost all scientific and artistic disciplines, in Music in particular, as is shown in this text. Group Theory extracts the essential characteristics of diverse situations in which some type of symmetry or transformation appears. In this text, a modern presentation of the fundamental notions of Group Theory is chosen, where the language of commutative diagrams and universal properties, so necessary in Modern Mathematics, in Physics and Computer Science, among other disciplines, is introduced.
Preface
The success of Group Theory is impressive and extraordinary. It is, perhaps, the most powerful and influential branch of all Mathematics. Its influence is strongly felt in almost all scientific and artistic disciplines (in Music, in particular) and in Mathematics itself. Group Theory extracts the essential characteristics of diverse situations in which some type of symmetry or transformation appears. Given a nonempty set, a binary operation is defined on it such that certain axioms hold, that is, it possesses a structure (the group structure). The concept of structure, and the concepts related to structure such as isomorphism, play a decisive role in modern Mathematics.
The general theory of structures is a powerful tool. Whenever someone proves that his objects of study satisfy the axioms of a certain structure, he immediately obtains all the valid results of the theory for his objects. There is no need to prove each one of the results in particular. Indeed, it can be said that the structures allow the classification of the different branches of Mathematics (or even the different objects in Music (! )).
The present text is based on the book in Spanish “Teoría de Grupos: un primer curso” by Emilio LluisPuebla, published by the Sociedad Matemática Mexicana This new text contains the material that corresponds to a course on the subject that is offered in the Mathematics Department of the Facultad de Ciencias of the Universidad Nacional Autónoma de México plus optional introductory material for a basic course on Mathematical Music Theory.
This text follows the approach of other texts by Emilio LluisPuebla on Linear Algebra and Homological Algebra. A modern presentation is chosen, where the language of commutative diagrams and universal properties, so necessary in Modern Mathematics, in Physics and Computer Science, among other disciplines, is introduced.
This work consists of four chapters. Each section contains a series of problems that can be solved with creativity by using the content that is presented there; these problems form a fundamental part of the text. They also are designed with the objective of reinforcing students’ mathematical writing. Throughout the first three chapters, representative examples (that are not numbered) of applications of Group Theory to Mathematical Music Theory are included for students who already have some knowledge of Music Theory.
In chapter 4, elaborated by Mariana Montiel, the application of Group Theory to Music Theory is presented in detail. Some basic aspects of Mathematical Music Theory are explained and, in the process, some essential elements of both areas are given to readers with different backgrounds. For this reason, the examples follow from some of the outstanding theoretical aspects of the previous chapters; the musical terms are introduced as they are needed so that a reader without musical background can understand the essence of how Group Theory is used to explain certain preestablished musical relations. On the other hand, for the reader with knowledge of Music Theory only, this chapter provides concrete elements, as well as motivation, to begin to understand Group Theory.
The last four authors give a special acknowledge for the valuable help in the English edition to Dr. Flor AceffSánchez who, in spite of her delicate health, put all her dedication and love in the elaboration of this text with many mathematical and musical comments. Without her, this text would never have come to life.
Content
 List of Figures
 Preface
 Introduction
 Chapter 1
 Binary Operations
 Algebraic Structures
 Elementary Properties
 Cyclic Groups
 Chapter 2
 Exact Sequences
 Quotient Groups
 Isomorphism Theorems
 Products
 Chapter 3
 Finitely Generated Abelian Groups
 Permutations, Orbits and Sylow Theorems
 Free Groups
 Tensor Product
 Chapter 4
 Musical Background
 The T and I Transformations
 The P, L and R Transformations
 The Isomorphism between PLR and TI
 The Duality of the TI and PLR Groups
 Solutions to the Problems of Chapter 4
 List of Symbols
 Index
 Bibliography and References
 The Authors
 Endnotes
About the Author
Flor AceffSánchez
Flor AceffSánchez obtained the Professor of Primary Education degree at the Escuela Nacional de Maestros in 1982. In 1989 obtained her undergraduate degree in Mathematics, in 1990 graduated as Master in Mathematics and obtained a Ph.D. in Mathematics in 1995 at the Faculty of Sciences of the Universidad Nacional Autónoma de México. She is currently Full Time Professor at the Faculty of Sciences of the Universidad Nacional Autónoma de México both in Professional and Posgraduate Studies. She was National Researcher (19951998).
She is member of various scientific associations such as: the Mexican Institute of Sciences and Humanities, the Royal Spanish Mathematical Society, the American Mathematical Society, the Mexican Society of Geography and Statistics, the Mathematical Society of Mexico and is president of the Council on Accreditation of Educational Programs in Mathematics.
Octavio A. AgustínAquino
Octavio Alberto Agustín Aquino studied his undergraduate degree in Applied Mathematics at the Universidad Tecnológica de la Mixteca in Huajuapan de León, Oaxaca, and the Masters in Mathematical Sciences at the Universidad Nacional Autónoma de México. He obtained his Ph.D. in 2011 at the Universidad Nacional Autónoma de México under the joint direction of Emilio LluisPuebla, Guerino Mazzola and Rodolfo San Agustín Chi. His research is aimed at extending the Mathematical Theory of Counterpoint developed by Guerino Mazzola.
Janine du Plessis
Janine du Plessis is originally from South Africa and did her undergraduate and Masters in Mathematics at Georgia State University, where she wrote her thesis under the direction of Mariana Montiel. She also studied Music at the same University. She presently teaches Mathematics at Georgia Perimeter College and Chattahoochee Technical College.
Emilio LluisPuebla
Emilio LluisPuebla completed his Professional and Master Studies in Mathematics in Mexico. In 1980 he obtained his Ph.D. in Mathematics in Canada. He is professor at the National Autonomous University of Mexico in their Professional and Graduate divisions for over thirty years. He has formed several professors and researchers who work in Mexico and abroad. His mathematical work has been established in his research articles published on Algebraic Ktheory and Cohomology of Groups in the most prestigious national and international journals. He has been Visiting Professor in Canada.
He received several academic awards, among others, Gabino Barreda Medal for the highest average in the Masters, National Researcher (19841990) and Endowed Chair of Excellence Conacyt (19921993). He is the author of several books on Algebraic KTheory, Homological Algebra, Linear Algebra and Mathematical Music Theory published worldwide by Addison Wesley, Springer Verlag, Birkhäuser, AMS, SMM, among others.
He is a member of several scientific associations such as the Royal Spanish Mathematical Society and the American Mathematical Society. He is president of the Academy of Sciences of the Mexican Institute of Sciences and Humanities, president of the Academy of Mathematics of the Mexican Society of Geography and Statistics and president 20002002 of the Mathematical Society of Mexico.
Mariana Montiel
Mariana Montiel did her undergraduate and Masters in Mathematics at the National Autonomous University of Mexico. In 2005 she obtained her Ph.D. in Mathematics in the United States. She is professor at Georgia State University, in the Department of Mathematics and Statistics, since 2006. Her research is focused on Mathematical Music Theory and Mathematics as a semiotic system, with emphasis in the aspect of language.
She has directed theses and research projects in Mexico and the United States in the area of Mathematical Music Theory. She has published articles and collaborated in books about the application of Category Theory to Mathematical Music Theory.
She has received several scholarships and grants from foundations and institutes in Mexico and the United States, such as the National Council of Science and Technology, The UNAM Foundation, the University of New Hampshire and the Research Foundation of Georgia State University. She collaborates internationally and does translations of articles on Mathematics and Mathematics Education. Some of her publications are found in Birkhäuser and Springer Verlag.