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GROUP THEORY Covers the University of Delhi B.Sc. Mathematics (Hons) syllabi of Algebra II (Sem-3) and Algebra V (Sem-6) Sudesh Kumari Shah Asha Gauri Shankar

Group Theory

Group Theory As per the University of Delhi B.Sc. Mathematics (Hons) syllabi for Algebra II (Sem-3) and Algebra V (Sem-6)

Sudesh Kumari Shah Associate Professor Sri Venkateswara College University of Delhi Asha Gauri Shankar Associate Professor Lakshmibai College University of Delhi

Assistant Editor–Acquisitions: Jigyasa Bhatia Associate Editor–Production: Sudipto Roy Copyright © 2013 Dorling Kindersley (India) Pvt. Ltd. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out or otherwise circulated without the publisher’s prior written consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser and without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the above-mentioned publisher of this book. ISBN: 978-81-317-8763-2 First Impression Second Impression, 2014 Published by Dorling Kindersley (India) Pvt. Ltd., lincencees of Pearson Education in South Asia. Head Office: Pearson Education, 7th Floor, Knowledge Boulevard, A-8 (A), Sector 62, Noida, India Registered Office: 11 Community Centre, Panchsheel Park, New Delhi, 110017, India. Compositor: County Caramels Printer:

A lot has changed from the Taj Mahal to the Delhi Metro, but what has not changed is that monumental feats always begin with strong pillars to stand on. Thank you Ma, Bauji, Pitaji, Bhai, Gauri Shankar, Keerti, Vinayak and Ajay for being my pillars. —Asha To my parents, Lalit, Akhil, Ritu, Tunnu, Tintin, Sumit, Swati and Chiya— for making everything worth it! —Sudesh

Foreword Abstract Algebra constitutes the algebraic structures of groups, rings, and fields. To understand Abstract Algebra, it is required to enter into the depth of each of these topics. I am very happy to see that this book is devoted entirely to groups with adequate depth so that it will help students to grasp the topic well and shall enable them to understand the other algebraic structures more easily. The book presents Group Theory in an easy and interesting manner to students. The authors have been successful in their attempt to explain the subject in a simple and lucid way. The learning objectives given at the beginning of each chapter will help the teacher to structure their lectures and enable students to monitor their own progress. The illustrations will help to understand the concepts. The stepby-step solutions will help students to master the problem-solving technique in an abstract setting. I am pleased to note that dihedral, quaternion, and Permutation groups have been dealt with in detail in a separate chapter as these serve handy as counter examples. A unique feature of the book is the chapter-wise summary of the frequently used results. The True/ False questions are the highlight of the book as they indicate the importance of the correct usage of certain terms in a statement. The answers to all exercises will help students to gain confidence while enhancing their problem solving abilities. It is my pleasure to write a foreword for this book written by Asha and Sudesh. My association with them has been for over 30 years. They have a thorough understanding of the subject and each of them has more than 30 years of teaching experience. I am sure that this book will be a ‘fun to read and solve while you learn’. I look forward to read more books written by Asha and Sudesh.      

B. K. Dass Professor and Head Department of Mathematics Dean, Faculty of Mathematical Sciences University of Delhi Delhi - 110007

Preface This book is devoted to algebraic groups which comprise the first essential step in the study of abstract algebra. Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, fields, vector spaces, modules, and algebras. The text includes a wide range of topics to cover diverse undergraduate curricula of group theory as taught at most universities. This book was conceptualized keeping the students in mind, with the aim of also serving both the newly initiated as well as the more advanced reader. While ensuring that the language remains easy to understand, we have not compromised with the rigour of the subject. Each topic has been illustrated with theorems, examples, and counter examples. To ensure that the book provides sufficient depth for the inquisitive reader, we have also included some topics that are taught in some universities at the postgraduate level. This conceptual range, from the most elementary to the relatively advanced topics, would also be helpful for students who are preparing for various competitive examinations. To ensure consistency, we have written each chapter along a defined sequence of sub-headings, starting with a brief outline of the chapter provided in learning objectives, definitions, theorems, solved problems, and exercises. All definitions and theorems are copiously illustrated with examples. We have noticed that many of our brightest students, who are comfortable with numerical problems, are sometimes uneasy with exercises in an abstract setting. Keeping this difficulty in mind and to demonstrate problem-solving techniques in an abstract setting, we have included several solved problems in each chapter. Exercises at the end of each section will also help active learning. For complicated or lengthy solutions, we have highlighted the individual steps and also provided an outline to clarify the underlying concept. In our teaching experience of more than three decades, we have noticed that most textbooks on abstract algebra do not provide answers to chapter-end exercises. This is a source of discomfort for most students, who would generally prefer to find these answers and confirm whether they did solve the exercises correctly! In this book, we have corrected this anomaly by including answers to all the exercises. For difficult problems, we have also provided hints to lead the reader to the correct solution. The idea is to make learning enjoyable and instill a sense of confidence in the reader. We suggest that the student should use the hints only after making a serious effort to solve the problems without any assistance. For the same reason, the answers and hints are provided separately at the end of the book so that these do not become a distraction. To emphasize the correct usage of certain important terms, we have included questions for which the reader is required to indicate whether it is true or false, with appropriate reasoning. Similarly, certain topics, which are of great importance for understanding and application, have been given the status of a chapter, to ensure that the student will devote sufficient time and also be able to easily refer to these. For instance, a separate chapter has been included on dihedral and quaternion groups, which are useful for providing examples and counter examples. Permutations have been divided into two chapters, where the first, Permutations I, deals with the basic concepts, and the second, Permutations II, was written to

Preface    ix cover the more advanced topics. A separate chapter covers the subgroups-centralizer, normalizer, and centre. A chapter-wise summary of results that are used frequently and should be remembered is given for ready reference. This will enable the student to prepare for examinations without having to search the library for separate texts which, at times, are hard to procure. We thank all of our students for their curiosity for sharing with us the difficulties they experienced in learning these topics and for helping us to refine this material over the years. We would also like to thank the reviewers of the manuscript for their valuable comments and suggestions. We convey our heartiest gratitude to our revered teachers, who sowed the seed of love in us for mathematics. A project like this takes a lot of time and effort. It would not have been possible to complete this work without the encouragement, guidance, help, and support of a lot of people. The list is too long to be enumerated. Our sincere and heartfelt thanks go to all of them. The production of a mathematics book is a very tedious process, and the editorial staff at Pearson Education has been very understanding, patient, and helpful. We convey our thanks to them. Finally, despite our very best efforts to check the accuracy, errors may still be left unnoticed. We would appreciate any comments, criticisms, questions or corrections from our readers. We hope that you will find this book enjoyable and useful.  

Sudesh Kumari Shah ([email protected]) Asha Gauri Shankar ([email protected])

Syllabus SEMESTER III Algebra II Symmetry of a square, dihedral groups, definitionand examples of groups including permutation groups and quaternion groups (illustration through matrices), elementary properties of groups, subgroups and examples of subgroups, centralizer, normalizer, centre of a group, cyclic groups, generators of cyclic groups, classificationof subgroups of cyclic groups, cycle notation for permutations, properties of per-mutations, even and odd permutations, alternating group, a check-digit scheme based on the dihedral group D5 , product (HK) of two subgroups, definitionand properties of cosets, Lagrange’s theorem and consequences including Fermat’s little theorem, an application of cosets to permutation groups, the rota-tion group of a cube and a soccer ball, definitionand examples of the external direct product of a finitenumber of groups, normal subgroups, factor groups, applications of factor groups to the alternating group A4, commutator subgroup, definitin and examples of homomorphism, properties of homomor-phism, definitio and examples of isomorphism, Cayley’s theorem, properties of isomorphism, Isomor-phism theorems I, II, and III, definitionand examples of automorphisms, inner automorphisms, auto-morphisms and inner automorphisms group, automorphism group of finiteand infinitecyclic groups, applications of factor groups to automorphisms groups, Cauchy’s theorem for finite abelian groups

SEMESTER VI Algebra V Properties of external direct products, the group of units modulo n as an external direct product, applications of external direct products to data security, public key cryptography, definitionand exam-ples of internal direct products, fundamental theorem of finiteabelian groups, definitionand examples of group actions, stabilizers and kernels of group actions, permutation representation associated with a given group action, applications of group actions: Cauchy’s theorem, index theorem, Cayley’s theorem, conjugacy relation, class equation and consequences, conjugacy in Sn, p-groups, Sylow’s theorems and consequences, definitionand examples of simple groups, nonsimplicity tests, composition series, Jordan-Holder theorem, solvable groups.

Contents* Foreword Preface Syllabus

1. Definition and Examples of Groups 1.1  Cayley Table 1.2  Groups of Numbers 1.3  Groups of Matrices 1.4  Groups of Functions 1.5  Group of Subsets of a Set 1.6  Group of Symmetries

2. Properties and Characterizations of Groups 2.1  Properties of Groups 2.2  Characterizations of Groups

3. Subgroups 3.1  Definition and Examples 3.2  Criteria for Subgroups 3.3  Some Subgroups of a Given Group 3.4  Intersection and Product of Subgroups 3.5  Subgroup Generated by a Subset

4. Cyclic Groups 4.1  Cyclic Subgroups 4.2  Cyclic Groups 4.3  Infinite Cyclic Groups 4.4  Finite Cyclic Groups 4.5  Subgroup Lattice *

Third semester: Chapters 1–14; Sixth semester: Chapters 14–24.

vii viii x

1 3 3 7 8 9 9

26 26 31

39 39 41 43 45 47

52 52 61 62 66 74

xii   Contents

5. Dihedral and Quaternion Groups 5.1  Dihedral Groups 5.2  Quaternion Group 5.3  Representation of Dihedral and Quarternion Groups as Matrices

6. Permutation Groups I

79 79 86 87


6.1  Symmetric Group Sn 92 6.2 Cycles 94 6.3  Alternating Group An 102 6.4  D8 as a Subgroup of S4 104

7. External Direct Products 7.1  External Direct Product of Two Groups 7.2  Properties of External Direct Products

8. Isomorphisms 8.1 Isomorphisms 8.2  Cayley’s Theorem

9. Lagrange’s Theorem

118 118 131

136 136

9.2  Lagrange’s Theorem 9.3  Application of Cosets to Permutation Groups

141 146

10. Normal Subgroups

11. Factor Groups 11.1  Definition and Examples 11.2  Properties of Factor Group 11.3  Commutator Subgroup

113 117

9.1 Cosets

10.1  Definition and Examples 10.2  Intersection and Product of Normal Subgroups


12. Homomorphisms 12.1  Definition and Examples 12.2  Kernel of a Homomorphism 12.3  Properties of Homomorphisms

154 154 156

163 163 166 173

178 178 180 185

Contents   xiii 12.4  Determining Homomorphisms of Cyclic Groups 12.5  Isomorphism Theorems

13. Automorphisms 13.1  Definition and Examples 13.2  Some Standard Automorphisms 13.3  Automorphism Group 13.4  Automorphism Groups of Cyclic Groups 13.5  Inner Automorphisms 13.6  Characteristic Subgroups

14. Centralizer Normalizer and Centre 14.1  Centralizer of an Element 14.2  Properties of Centralizer of an Element 14.3  Centralizer of a Subset 14.4  Centre of a Group 14.5  Normalizer of a Subset

15. Direct Products 15.1  Properties of External Direct Products 15.2  U(n) as External Direct Product 15.3  Internal Direct Products

16. Fundamental Theorem of Finite Abelian Groups 16.1  Fundamental Theorem 16.2  Applications of Fundamental Theorem of Finite Abelian Groups

17. Group Action 17.1  Definition and Examples 17.2  Orbits of a Group Action

18. Permutation Representation of a Group 18.1  Definition and Examples

188 195

204 204 205 206 209 211 215

220 220 221 222 223 224

231 231 240 245

253 254 259

267 267 270

276 276

19. Generalized Cayley’s Theorem


20. Conjugacy and Class Equation


21. Permutation Groups II


21.1  Cyclic Decomposition 21.2  Conjugacy in Sn

294 295

xiv   Contents 21.3  Generators of Sn and An 298 21.4  Simplicity of An 299

22. Sylow’s Theorems and Non-Simplicity Tests 22.1  Sylow’s Theorems 22.2  Non-simplicity Tests

23. Composition Series 23.1  Subnormal and Normal Series 23.2  Composition Series 23.3  Jordan–Hölder Theorem

24. Solvable Groups 24.1  Definition and Examples 24.2  Equivalent Conditions of Solvability 24.3  Higher Commutator Subgroups

309 309 322

328 328 329 331

339 339 343 346

Summary of Important Results 351 Answers and Hints 367 Index 396 Notations Used 400

1 Definition and Examples of Groups Learning Objectives In this chapter, we shall  define

 give

 study

 study

various algebraic structures like groupoid, semi-group, monoid, and group the relationships between these algebraic structures

examples of these from numbers, matrices, and functions the symmetries of a square and a rectangle

In this chapter, we will define a group and give numerous examples, which will be used throughout the book. Therefore we are devoting a considerable amount of time on them. This chapter will also be very useful for ready ­reference. A non-empty set equipped with one or more binary operations is called an algebraic structure. There are several algebraic structures with one binary operation and for the sake of completeness we shall define them below. Definition 1.1  Let S be a non-empty set. A mapping *: S × S → S is called a binary operation on S. The image of (s, t) ∈ S × S under the operation * is written as s * t. ♦ If * is a binary operation on a set S, then we also say that S is closed with respect to * Notation  The system consisting of a non-empty set G together with a binary operation * defined on G is denoted by (G, * ). Definition 1.2  Let G be a non-empty set and * a binary operation on G. (i) Then (G, *) is called a groupoid. (ii) A groupoid (G, *) in which * is associative is called a semi-group.

GROUP THEORY Covers the University of Delhi B.Sc. Mathematics (Hons) syllabi of Algebra II (Sem-3) and Algebra V (Sem-6) Sudesh Kumari Shah | Asha Gauri Shankar Designed as a text for undergraduate students of mathematics, this book meets the curriculum requirements of the University of Delhi. It specifically caters to the syllabus of group theory covered in Algebra II (Semester 3) and Algebra V (Semester 6) papers and follows an easy-paced treatment of the basic concepts. SALIENT FEATURES w Learning objectives at the beginning of each chapter w Concepts illustrated with 550 solved examples w Stepwise proofs of 200 theorems w Chapter-wise summary for ready reference COVERAGE z Definition and Examples of Groups z Properties and Characterizations of Groups z Subgroups z Cyclic Groups z Dihedral and Quarternion Groups z Permutation Groups I z External Direct Products z Isomorphisms z Lagrange’s Theorem z Normal Subgroups z Factor Groups z Homomorphisms

w w w

z z z z z z z z z z z z

Graded exercises Concepts reinforced by true/false questions Answers/hints to all the 540 unsolved problems

Automorphisms Centralizer Normalizer and Centre Direct Products Fundamental Theorem of Finite Abelian Groups Group Action Permutation Representation of a Group Generalized Cayley’s Theorem Conjugacy and Class Equation Permutation Groups II Sylow’s Theorems and Non-Simplicity Tests Composition Series Solvable Groups

Dr Sudesh Kumari Shah is Associate Professor at the Department of Mathematics, Sri Venkateswara College, University of Delhi. An alumnus of IIT Delhi, she has been teaching undergraduate and postgraduate students of the University of Delhi for more than 30 years. Dr Asha Gauri Shankar is Associate Professor at the Department of Mathematics, Lakshmibai College, University of Delhi. She obtained a Ph.D. from Imperial College London. She was awarded 'Teacher of Excellence' in 2009 by the University of Delhi. The award was conferred by Dr A. P. J. Abdul Kalam.

ISBN 978-81-317-8763-2

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