# elements of abstract algebra

(Parte 1 de 9)

Elements of Abstract and Linear Algebra

E. H. Connell

E.H. Connell Department of Mathematics University of Miami P.O. Box 249085 Coral Gables, Florida 33124 USA ec@math.miami.edu

Introduction

In 1965 I rst taught an undergraduate course in abstract algebra. It was fun to teach because the material was interesting and the class was outstanding. Five of those students later earned a Ph.D. in mathematics. Since then I have taught the course about a dozen times from various texts. Over the years I developed a set of lecture notes and in 1985 I had them typed so they could be used as a text. They now appear (in modi ed form) as the rst ve chapters of this book. Here were some of my motives at the time.

1) To have something as short and inexpensive as possible. In my experience, students like short books.

2) To avoid all innovation. To organize the material in the most simple-minded straightforward manner.

3) To order the material linearly. To the extent possible, each section should use the previous sections and be used in the following sections.

4) To omit as many topics as possible. This is a foundational course, not a topics course. If a topic is not used later, it should not be included. There are three good reasons for this. First, linear algebra has top priority. It is better to go forward and do more linear algebra than to stop and do more group and ring theory. Second, it is more important that students learn to organize and write proofs themselves than to cover more subject matter. Algebra is a perfect place to get started because there are many \easy" theorems to prove. There are many routine theorems stated here without proofs, and they may be considered as exercises for the students. Third, the material should be so fundamental that it be appropriate for students in the physical sciences and in computer science. Zillions of students take calculus and cookbook linear algebra, but few take abstract algebra courses. Something is wrong here, and one thing wrong is that the courses try to do too much group and ring theory and not enough matrix theory and linear algebra.

5) To o er an alternative for computer science majors to the standard discrete mathematics courses. Most of the material in the rst four chapters of this text is covered in various discrete mathematics courses. Computer science majors might bene t by seeing this material organized from a purely mathematical viewpoint.

Over the years I used the ve chapters that were typed as a base for my algebra courses, supplementing them as I saw t. In 1996 I wrote a sixth chapter, giving enough material for a full rst year graduate course. This chapter was written in the same \style" as the previous chapters, i.e., everything was right down to the nub. It hung together pretty well except for the last two sections on determinants and dual spaces. These were independent topics stuck on at the end. In the academic year 1997-98 I revised all six chapters and had them typed in LaTeX. This is the personal background of how this book came about.

It is di cult to do anything in life without help from friends, and many of my friends have contributed to this text. My sincere gratitude goes especially to Marilyn Gonzalez, Lourdes Robles, Marta Alpar, John Zweibel, Dmitry Gokhman, Brian Coomes, Huseyin Kocak, and Shulim Kaliman. To these and all who contributed, this book is fondly dedicated.

This book is a survey of abstract algebra with emphasis on linear algebra. It is intended for students in mathematics, computer science, and the physical sciences. The rst three or four chapters can stand alone as a one semester course in abstract algebra. However they are structured to provide the background for the chapter on linear algebra. Chapter 2 is the most di cult part of the book because groups are written in additive and multiplicative notation, and the concept of coset is confusing at rst. After Chapter 2 the book gets easier as you go along. Indeed, after the rst four chapters, the linear algebra follows easily. Finishing the chapter on linear algebra gives a basic one year undergraduate course in abstract algebra. Chapter 6 continues the material to complete a rst year graduate course. Classes with little background can do the rst three chapters in the rst semester, and chapters 4 and 5 in the second semester. More advanced classes can do four chapters the rst semester and chapters 5 and 6 the second semester. As bare as the rst four chapters are, you still have to truck right along to nish them in one semester.

The presentation is compact and tightly organized, but still somewhat informal.

The proofs of many of the elementary theorems are omitted. These proofs are to be provided by the professor in class or assigned as homework exercises. There is a non-trivial theorem stated without proof in Chapter 4, namely the determinant of the product is the product of the determinants. For the proper ow of the course, this theorem should be assumed there without proof. The proof is contained in Chapter 6. The Jordan form should not be considered part of Chapter 5. It is stated there only as a reference for undergraduate courses. Finally, Chapter 6 is not written primarily for reference, but as an additional chapter for more advanced courses.

This text is written with the conviction that it is more e ective to teach abstract and linear algebra as one coherent discipline rather than as two separate ones. Teaching abstract algebra and linear algebra as distinct courses results in a loss of synergy and a loss of momentum. Also with this text the professor does not extract the course from the text, but rather builds the course upon it. I am convinced it is easier to build a course from a base than to extract it from a big book. Because after you extract it, you still have to build it. The bare bones nature of this book adds to its exibility, because you can build whatever course you want around it. Basic algebra is a subject of incredible elegance and utility, but it requires a lot of organization. This book is my attempt at that organization. Every e ort has been extended to make the subject move rapidly and to make the ow from one topic to the next as seamless as possible. The student has limited time during the semester for serious study, and this time should be allocated with care. The professor picks which topics to assign for serious study and which ones to \wave arms at". The goal is to stay focused and go forward, because mathematics is learned in hindsight. I would have made the book shorter, but I did not have any more time.

When using this text, the student already has the outline of the next lecture, and each assignment should include the study of the next few pages. Study forward, not just back. A few minutes of preparation does wonders to leverage classroom learning, and this book is intended to be used in that manner. The purpose of class is to learn, not to do transcription work. When students come to class cold and spend the period taking notes, they participate little and learn little. This leads to a dead class and also to the bad psychology of \OK, I am here, so teach me the subject." Mathematics is not taught, it is learned, and many students never learn how to learn. Professors should give more direction in that regard.

Unfortunately mathematics is a di cult and heavy subject. The style and approach of this book is to make it a little lighter. This book works best when viewed lightly and read as a story. I hope the students and professors who try it, enjoy it.

E. H. Connell

Department of Mathematics University of Miami Coral Gables, FL 33124 ec@math.miami.edu

Outline

Chapter 1 Background and Fundamentals of Mathematics

Sets, Cartesian products 1 Relations, partial orderings, Hausdor maximality principle, 3 equivalence relations

Functions, bijections, strips, solutions of equations, 5 right and left inverses, projections

Notation for the logic of mathematics 13 Integers, subgroups, unique factorization 14

Chapter 2 Groups

Groups, scalar multiplication for additive groups 19 Subgroups, order, cosets 21 Normal subgroups, quotient groups, the integers mod n 25 Homomorphisms 27 Permutations, the symmetric groups 31 Product of groups 34

Chapter 3 Rings

Rings 37 Units, domains, elds 38 The integers mod n 40 Ideals and quotient rings 41 Homomorphisms 42 Polynomial rings 45 Product of rings 49 The Chinese remainder theorem 50 Characteristic 50 Boolean rings 51

Chapter 4 Matrices and Matrix Rings

Addition and multiplication of matrices, invertible matrices 53 Transpose 56 Triangular, diagonal, and scalar matrices 56 Elementary operations and elementary matrices 57 Systems of equations 59 vii

Determinants, the classical adjoint 60 Similarity, trace, and characteristic polynomial 64

Chapter 5 Linear Algebra

Modules, submodules 68 Homomorphisms 69 Homomorphisms on Rn 71 Cosets and quotient modules 74 Products and coproducts 75 Summands 7 Independence, generating sets, and free basis 78 Characterization of free modules 79 Uniqueness of dimension 82 Change of basis 83 Vector spaces, square matrices over elds, rank of a matrix 85 Geometric interpretation of determinant 90 Linear functions approximate di erentiable functions locally 91 The transpose principle 92 Nilpotent homomorphisms 93 Eigenvalues, characteristic roots 95 Jordan canonical form 96 Inner product spaces, Gram-Schmidt orthonormalization 98 Orthogonal matrices, the orthogonal group 102 Diagonalization of symmetric matrices 103

Chapter 6 Appendix

The Chinese remainder theorem 108 Prime and maximal ideals and UFDs 109 Splitting short exact sequences 114 Euclidean domains 116 Jordan blocks 122 Jordan canonical form 123 Determinants 128 Dual spaces 130 viii

Abstract algebra is not only a major subject of science, but it is also magic and fun. Abstract algebra is not all work and no play, and it is certainly not a dull boy. See, for example, the neat card trick on page 18. This trick is based, not on sleight of hand, but rather on a theorem in abstract algebra. Anyone can do it, but to understand it you need some group theory. And before beginning the course, you might rst try your skills on the famous (some would say infamous) tile puzzle. In this puzzle, a frame has 12 spaces, the rst 1 with numbered tiles and the last vacant. The last two tiles are out of order. Is it possible to slide the tiles around to get them all in order, and end again with the last space vacant? After giving up on this, you can study permutation groups and learn the answer!

Chapter 1

Background and Fundamentals of Mathematics

This chapter is fundamental, not just for algebra, but for all elds related to mathematics. The basic concepts are products of sets, partial orderings, equivalence relations, functions, and the integers. An equivalence relation on a set A is shown to be simply a partition of A into disjoint subsets. There is an emphasis on the concept of function, and the properties of surjective, injective, and bijective. The notion of a solution of an equation is central in mathematics, and most properties of functions can be stated in terms of solutions of equations. In elementary courses the section on the Hausdor Maximality Principle should be ignored. The nal section gives a proof of the unique factorization theorem for the integers.

Notation Mathematics has its own universally accepted shorthand. The symbol ∃ means \there exists" and 9! means \there exists a unique". The symbol 8 means \for each" and ) means \implies". Some sets (or collections) are so basic they have their own proprietary symbols. Five of these are listed below.

 Sets Suppose A;B;C; are sets. We use the standard notation for intersection

and union.

A \ B = fx : x 2 A and x 2 Bg = the set of all x which are elements

2 Background Chapter 1 of A and B.

A ∪ B = fx : x 2 A or x 2 Bg = the set of all x which are elements of A or B.

Any set called an index set is assumed to be non-void. Suppose T is an index set and for each t 2 T, At is a set.

Let ; be the null set. If A \ B = ;, then A and B are said to be disjoint.

De nition Suppose each of A and B is a set. The statement that A is a subset of B (A B) means that if a is an element of A, then a is an element of B. That is, a 2 A ) a 2 B: If A B we may say A is contained in B, or B contains A.

Exercise Suppose each of A and B is a set. The statement that A is not a subset of B means .

Theorem (De Morgan’s laws) Suppose S is a set. If C S (i.e., if C is a subset of S), let C0, the complement of C in S, be de ned by C0 = S C = fx 2 S : x 62 Cg. Then for any A;B S,

(Parte 1 de 9)