MMT-003 ALGEBRA Indira Gandhi National Open University School of Sciences Block 1 GROUP THEORY Course Introduction 3 Block Introduction 5 Notations and Symbols 6 UNIT 1 Basic Group Theory – A Review 7 UNIT 2 Group Actions 31 UNIT 3 The Sylow Theorems 57 Course Design Committee* Prof. Gurmeet Kaur Bakshi Prof. Parvati Shastri Deptt. of Mathmatics, Centre for Excellence in Basic Sciences Panjab University Mumbai University Dr. Anuj Bishnoi Faculty members Deptt. of Mathematics, School of Sciences, IGNOU University of Delhi Dr. Deepika Prof. K. N. Raghavan Prof. Poornima Mital Deptt. of Mathematics Prof. Parvin Sinclair Institute of Mathematical Sciences, Chennai Prof. Sujatha Varma Prof. Ravi Rao Dr. S. Venkataraman School of Mathematics Tata Institute of Fundamental Research Mumbai * The course design is based on the recommendations of the Expert Committee for the programme M.Sc (Mathematics with Applications in Computer Science). Block Preparation Team Prof. B. Sury (Editor) Prof. Parvin Sinclair Deptt. of Mathematics School of Sciences Indian Statistical Institute, Bengaluru IGNOU Prof. K. N. Raghavan Deptt. of Maths Institute of Mathematical Sciences, Chennai Course Coordinator: Prof. Parvin Sinclair Acknowledgement: To Sh. S. S. Chauhan for the preparation of the CRC of this block. November, 2018 © Indira Gandhi National Open University ISBN-8]- All right reserved. No part of this work may be reproduced in any form, by mimeograph or any other means, without permission in writing from the Indira Gandhi National Open University. Further information on the Indira Gandhi National Open University courses, may be obtained from the University’s office at Maidan Garhi, New Delhi-110 068 and IGNOU website www.ignou.ac.in. Printed and published on behalf of the Indira Gandhi National Open University, New Delhi by 2 Prof. M. S. Nathawat, School of Sciences. COURSE INTRODUCTION Welcome to the next level of your study of algebra, the previous one being the course on Linear Algebra. As you know by now, abstract algebra is the single vehicle that encapsulates all the aspects of mathematical thinking — particularisation and generalisation, abstraction, brevity and clarity of expression, precision in communicating proofs. Through this course we aim to help you develop these abilities and processes further, while studying some abstract concepts in group, ring and field theory. Of course, you may not find this to be a “standard” algebra course. It has, actually, been specially designed to help you develop a better understanding of some of the application-oriented courses in the next two semesters of this Master’s programme. Therefore, you will find that the concepts and processes that you study in this will be closely tied to examples of their applications alongside. Before getting further into the details of this course, here is a brief historical overview. The origins of group theory lie in the theory of substitutions, developed by the prolific 18th and 19th century mathematicians Lagrange and Galois, whose work you will keep meeting in this course. Interestingly, while most courses on algebra, including this one, begin with group theory, it is the theory of rings that began developing earlier. Among the abstract notions of ring theory, the concrete example we first see is that of the ring of integers. Natural generalisations like the ring of integers modulo n and the ring of Gaussian integers occurred in Gauss's work and, simultaneously, rings of polynomials also arose. Initially, they were studied independently, with the former having applications to number theory and the latter giving tools to study geometric problems. Soon, it was realized that it would be beneficial to unify them under the same umbrella of abstract rings. The abstract notion of a commutative ring was first given by Fraenkel in 1914. However, it was the algebraist Emmy Noether, considered by many as the mother of modern algebra, who extended and abstracted the theories of polynomial rings to create abstract commutative ring theory in the 1920s. (A major part of ring theory is non-commutative ring theory, which we shall not really be dealing with in this course.) You are familiar with the fields of real/complex numbers. In fact, field theory (especially finite fields) came up in Galois's work in the theory of equations. However, it is Weber who gave the first clear definition of an abstract field, in 1893. Later, in 1910, Ernst Steinitz (1871-1928) synthesised the knowledge of abstract field theory accumulated till then. He axiomatically studied the properties of fields and clearly defined many important field-theoretic concepts which you will be studying in Block 4 of this course. In modern times, finite fields have found many applications to areas like coding theory and cryptography. You will study some of these in this course, and details of the applications in the relevant courses in the coming semesters. Now let us present how the course unfolds. There are two main paths through the land of groups. One emphasises the action of groups on sets, graphs and geometries. Here generators and relations play a central role and, in the finite case, counting arguments come to the fore. The other emphasises the action of groups on vector spaces and modules, both internal and external. Here linear algebra is the key tool. Sometimes these two paths intersect fruitfully. Often they go their divergent ways. The groups that act ‘linearly’ are particularly important and powerful. You will study groups from the first path’s point of view in Block 1, and in Block 2 you will get a flavour of the second path too. 3 In Block 3, we focus on basics of ring theory, some of which will be a recalling of what you would have studied earlier. We also discuss congruences here, which has applications in different areas like coding theory and cryptography. In Block 4, you will be studying fields, with the focus on finite fields and their applications. You will also study some Galois theory here. Now, a word about our notation. Each unit is divided into sections, which may be further divided into sub-sections. These sections/sub-sections are numbered sequentially, as are the exercises and important equations, within a unit. Regarding the exercises in each unit, these have been placed at many points in a unit. They are meant for you to assess if you have understood the concepts and processes that have been discussed till that point. Therefore, you must attempt to solve all the exercises in the unit as you go along, before you look up the solution. Since the material in the different units is heavily interlinked, there will be cross- referencing. For this we will be using the notation Sec.x.y to mean Section y of Unit x. Further, the end of an example is shown by the sign ∗∗∗ and the end of a proof of a theorem/proposition/corollary is shown by the mark . Another compulsory component of this course is its assignment, which covers the whole course. Your academic counsellor at the study centre will evaluate it, and return it to you with detailed comments. Thus, the assignment is meant to be a teaching as well as an assessment aid. Further, you will not be allowed to take the exam of this course till you submit your assignment response. So please submit it well in time. The course material that we have sent you is self-sufficient. If you have a problem in understanding any portion, please ask your academic counsellor for help. In addition to these exercises, you may like to also look up some other books in the library of your study centre, and try to solve some exercises from these books too. Some useful books and websites are the following: 1) “Algebra”, M. Artin, Pearson 2) “Abstract Algebra”, D.S.Dummit and R.M.Foote, Wiley (Student Edition) 3) “Contemporary Abstract Algebra”, J. Gallian, Narosa 4) “Abstract Algebra: Theory and Applications”, T. W. Judson (freely downloadable e-book) 5) https://www.youtube.com/watch?v=ylAXYqgbp4M (on group theory) 6) http://www.jmilne.org/math/CourseNotes/GT.pdf 7) http://www.math.uconn.edu/~kconrad/blurbs/ Best wishes for an enjoyable time with algebra! The Course Team 4 BLOCK INTRODUCTION There are basically two groups of people – those who act, and those who claim to act. The first group is less crowded! Paraphrasing Mark Twain With this block you will resume your study of group theory. This area of mathematics has many applications. Be it a combinatorial problem like counting numbers of isomers of a chemical compound, or the problem of whether a polynomial in one variable admits its roots to be expressed in terms of the coefficients if we allow only algebraic operations and taking of square, cube and higher roots, finite groups play a crucial role. This is not surprising as the conceptual structure behind any kind of symmetry is that of a group. To start with, in Unit 1 we will help you recall the basic concepts that you studied in your undergraduate studies. So, we will briefly review definitions and results about cosets, normal subgroups, quotient groups, etc. Unit 2 is focused on group actions. This concept is the reason behind the importance of groups in such diverse areas of science and mathematics. Here you will study about ‘defining actions’, ‘natural actions’ and several other kinds of actions. Related to an action are the concepts of an ‘orbit’ and a ‘stabiliser’. You will study these objects, their properties and the relationship between them. You will also study about the Class Equation here. In Unit 3 you will study how group actions can be used to prove some very fundamental results about finite groups.