abstract algebra notes for study, Lecture notes of Algebra

Download abstract algebra notes for study and more Lecture notes Algebra in PDF only on Docsity!

Abstract Algebra

Theory and Applications

Thomas W. Judson

Stephen F. Austin State University

August 27, 2010

ii

Copyright 1997 by Thomas W. Judson.

Permission is granted to copy, distribute and/or modify this document un- der the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invari- ant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix entitled “GNU Free Documentation License”.

A current version can always be found via abstract.pugetsound.edu.

iv PREFACE

hand, if applications are to be emphasized, the course might cover Chapters 1 through 14, and 16 through 22. In an applied course, some of the more the- oretical results could be assumed or omitted. A chapter dependency chart appears below. (A broken line indicates a partial dependency.)

Chapter 23

Chapter 22

Chapter 21

Chapter 18 Chapter 20 Chapter 19

Chapter 17 Chapter 15

Chapter 13 Chapter 16 Chapter 12 Chapter 14

Chapter 11

Chapter 10

Chapter 8 Chapter 9 Chapter 7

Chapters 1–

Though there are no specific prerequisites for a course in abstract alge- bra, students who have had other higher-level courses in mathematics will generally be more prepared than those who have not, because they will pos- sess a bit more mathematical sophistication. Occasionally, we shall assume some basic linear algebra; that is, we shall take for granted an elemen- tary knowledge of matrices and determinants. This should present no great problem, since most students taking a course in abstract algebra have been introduced to matrices and determinants elsewhere in their career, if they have not already taken a sophomore- or junior-level course in linear algebra.

PREFACE v

Exercise sections are the heart of any mathematics text. An exercise set appears at the end of each chapter. The nature of the exercises ranges over several categories; computational, conceptual, and theoretical problems are included. A section presenting hints and solutions to many of the exercises appears at the end of the text. Often in the solutions a proof is only sketched, and it is up to the student to provide the details. The exercises range in difficulty from very easy to very challenging. Many of the more substantial problems require careful thought, so the student should not be discouraged if the solution is not forthcoming after a few minutes of work. There are additional exercises or computer projects at the ends of many of the chapters. The computer projects usually require a knowledge of pro- gramming. All of these exercises and projects are more substantial in nature and allow the exploration of new results and theory.

Acknowledgements

I would like to acknowledge the following reviewers for their helpful com- ments and suggestions.

• David Anderson, University of Tennessee, Knoxville
• Robert Beezer, University of Puget Sound
• Myron Hood, California Polytechnic State University
• Herbert Kasube, Bradley University
• John Kurtzke, University of Portland
• Inessa Levi, University of Louisville
• Geoffrey Mason, University of California, Santa Cruz
• Bruce Mericle, Mankato State University
• Kimmo Rosenthal, Union College
• Mark Teply, University of Wisconsin

I would also like to thank Steve Quigley, Marnie Pommett, Cathie Griffin, Kelle Karshick, and the rest of the staff at PWS for their guidance through- out this project. It has been a pleasure to work with them.

Thomas W. Judson

• 1 Preliminaries Preface iii
  • 1.1 A Short Note on Proofs
  • 1.2 Sets and Equivalence Relations
• 2 The Integers
  • 2.1 Mathematical Induction
  • 2.2 The Division Algorithm
• 3 Groups
  • 3.1 The Integers mod n and Symmetries
  • 3.2 Definitions and Examples
  • 3.3 Subgroups
• 4 Cyclic Groups
  • 4.1 Cyclic Subgroups
  • 4.2 The Group C∗
  • 4.3 The Method of Repeated Squares
• 5 Permutation Groups
  • 5.1 Definitions and Notation
  • 5.2 The Dihedral Groups
• 6 Cosets and Lagrange’s Theorem
  • 6.1 Cosets
  • 6.2 Lagrange’s Theorem
  • 6.3 Fermat’s and Euler’s Theorems
• 7 Introduction to Cryptography CONTENTS vii
  • 7.1 Private Key Cryptography
  • 7.2 Public Key Cryptography
• 8 Algebraic Coding Theory
  • 8.1 Error-Detecting and Correcting Codes
  • 8.2 Linear Codes
  • 8.3 Parity-Check and Generator Matrices
  • 8.4 Efficient Decoding
• 9 Isomorphisms
  • 9.1 Definition and Examples
  • 9.2 Direct Products
• 10 Normal Subgroups and Factor Groups
  • 10.1 Factor Groups and Normal Subgroups
  • 10.2 Simplicity of An
• 11 Homomorphisms
  • 11.1 Group Homomorphisms
  • 11.2 The Isomorphism Theorems
• 12 Matrix Groups and Symmetry
  • 12.1 Matrix Groups
  • 12.2 Symmetry
• 13 The Structure of Groups
  • 13.1 Finite Abelian Groups
  • 13.2 Solvable Groups
• 14 Group Actions
  • 14.1 Groups Acting on Sets
  • 14.2 The Class Equation
  • 14.3 Burnside’s Counting Theorem
• 15 The Sylow Theorems
  • 15.1 The Sylow Theorems
  • 15.2 Examples and Applications
• 16 Rings viii CONTENTS
  • 16.1 Rings
  • 16.2 Integral Domains and Fields
  • 16.3 Ring Homomorphisms and Ideals
  • 16.4 Maximal and Prime Ideals
  • 16.5 An Application to Software Design
• 17 Polynomials
  • 17.1 Polynomial Rings
  • 17.2 The Division Algorithm
  • 17.3 Irreducible Polynomials
• 18 Integral Domains
  • 18.1 Fields of Fractions
  • 18.2 Factorization in Integral Domains
• 19 Lattices and Boolean Algebras
  • 19.1 Lattices
  • 19.2 Boolean Algebras
  • 19.3 The Algebra of Electrical Circuits
• 20 Vector Spaces
  • 20.1 Definitions and Examples
  • 20.2 Subspaces
  • 20.3 Linear Independence
• 21 Fields
  • 21.1 Extension Fields
  • 21.2 Splitting Fields
  • 21.3 Geometric Constructions
• 22 Finite Fields
  • 22.1 Structure of a Finite Field
  • 22.2 Polynomial Codes
• 23 Galois Theory
  • 23.1 Field Automorphisms
  • 23.2 The Fundamental Theorem
  • 23.3 Applications
• Hints and Solutions

x CONTENTS

Preliminaries

A certain amount of mathematical maturity is necessary to find and study applications of abstract algebra. A basic knowledge of set theory, mathe- matical induction, equivalence relations, and matrices is a must. Even more important is the ability to read and understand mathematical proofs. In this chapter we will outline the background needed for a course in abstract algebra.

1.1 A Short Note on Proofs

Abstract mathematics is different from other sciences. In laboratory sciences such as chemistry and physics, scientists perform experiments to discover new principles and verify theories. Although mathematics is often motivated by physical experimentation or by computer simulations, it is made rigorous through the use of logical arguments. In studying abstract mathematics, we take what is called an axiomatic approach; that is, we take a collection of objects S and assume some rules about their structure. These rules are called axioms. Using the axioms for S, we wish to derive other information about S by using logical arguments. We require that our axioms be consistent; that is, they should not contradict one another. We also demand that there not be too many axioms. If a system of axioms is too restrictive, there will be few examples of the mathematical structure. A statement in logic or mathematics is an assertion that is either true or false. Consider the following examples:

• 3 + 56 − 13 + 8/2.
• All cats are black.
• 2 + 3 = 5.

1

1.1 A SHORT NOTE ON PROOFS 3

ax^2 + bx + c = 0 with a 6 = 0 is true, then the conclusion must be true. A proof of this statement might simply be a series of equations:

ax^2 + bx + c = 0

x^2 +

b a x = −

c a

x^2 + b a x +

b 2 a

b 2 a

c a ( x + b 2 a

b^2 − 4 ac 4 a^2

x + b 2 a

b^2 − 4 ac 2 a

x =

−b ±

b^2 − 4 ac 2 a

If we can prove a statement true, then that statement is called a propo- sition. A proposition of major importance is called a theorem. Sometimes instead of proving a theorem or proposition all at once, we break the proof down into modules; that is, we prove several supporting propositions, which are called lemmas, and use the results of these propositions to prove the main result. If we can prove a proposition or a theorem, we will often, with very little effort, be able to derive other related propositions called corollaries.

Some Cautions and Suggestions

There are several different strategies for proving propositions. In addition to using different methods of proof, students often make some common mis- takes when they are first learning how to prove theorems. To aid students who are studying abstract mathematics for the first time, we list here some of the difficulties that they may encounter and some of the strategies of proof available to them. It is a good idea to keep referring back to this list as a reminder. (Other techniques of proof will become apparent throughout this chapter and the remainder of the text.)

• A theorem cannot be proved by example; however, the standard way to show that a statement is not a theorem is to provide a counterexample.
• Quantifiers are important. Words and phrases such as only, for all, for every, and for some possess different meanings.

4 CHAPTER 1 PRELIMINARIES

• Never assume any hypothesis that is not explicitly stated in the theo- rem. You cannot take things for granted.
• Suppose you wish to show that an object exists and is unique. First show that there actually is such an object. To show that it is unique, assume that there are two such objects, say r and s, and then show that r = s.
• Sometimes it is easier to prove the contrapositive of a statement. Prov- ing the statement “If p, then q” is exactly the same as proving the statement “If not q, then not p.”
• Although it is usually better to find a direct proof of a theorem, this task can sometimes be difficult. It may be easier to assume that the theorem that you are trying to prove is false, and to hope that in the course of your argument you are forced to make some statement that cannot possibly be true. Remember that one of the main objectives of higher mathematics is proving theorems. Theorems are tools that make new and productive ap- plications of mathematics possible. We use examples to give insight into existing theorems and to foster intuitions as to what new theorems might be true. Applications, examples, and proofs are tightly interconnected— much more so than they may seem at first appearance.

1.2 Sets and Equivalence Relations

Set Theory

A set is a well-defined collection of objects; that is, it is defined in such a manner that we can determine for any given object x whether or not x belongs to the set. The objects that belong to a set are called its elements or members. We will denote sets by capital letters, such as A or X; if a is an element of the set A, we write a ∈ A. A set is usually specified either by listing all of its elements inside a pair of braces or by stating the property that determines whether or not an object x belongs to the set. We might write

X = {x 1 , x 2 ,... , xn}

for a set containing elements x 1 , x 2 ,... , xn or

X = {x : x satisfies P}

6 CHAPTER 1 PRELIMINARIES

We can consider the union and the intersection of more than two sets. In this case we write (^) n ⋃

i=

Ai = A 1 ∪... ∪ An

and (^) n ⋂

i=

Ai = A 1 ∩... ∩ An

for the union and intersection, respectively, of the collection of sets A 1 ,... An. When two sets have no elements in common, they are said to be disjoint; for example, if E is the set of even integers and O is the set of odd integers, then E and O are disjoint. Two sets A and B are disjoint exactly when A ∩ B = ∅. Sometimes we will work within one fixed set U , called the universal set. For any set A ⊂ U , we define the complement of A, denoted by A′, to be the set A′^ = {x : x ∈ U and x /∈ A}. We define the difference of two sets A and B to be

A \ B = A ∩ B′^ = {x : x ∈ A and x /∈ B}.

Example 1. Let R be the universal set and suppose that

A = {x ∈ R : 0 < x ≤ 3 } and B = {x ∈ R : 2 ≤ x < 4 }.

Then

A ∩ B = {x ∈ R : 2 ≤ x ≤ 3 } A ∪ B = {x ∈ R : 0 < x < 4 } A \ B = {x ∈ R : 0 < x < 2 } A′^ = {x ∈ R : x ≤ 0 or x > 3 }.



Proposition 1.1 Let A, B, and C be sets. Then

1. A ∪ A = A, A ∩ A = A, and A \ A = ∅;
2. A ∪ ∅ = A and A ∩ ∅ = ∅;

1.2 SETS AND EQUIVALENCE RELATIONS 7

3. A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C;
4. A ∪ B = B ∪ A and A ∩ B = B ∩ A;
5. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C);
6. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Proof. We will prove (1) and (3) and leave the remaining results to be proven in the exercises. (1) Observe that

A ∪ A = {x : x ∈ A or x ∈ A} = {x : x ∈ A} = A

and

A ∩ A = {x : x ∈ A and x ∈ A} = {x : x ∈ A} = A.

Also, A \ A = A ∩ A′^ = ∅. (3) For sets A, B, and C,

A ∪ (B ∪ C) = A ∪ {x : x ∈ B or x ∈ C} = {x : x ∈ A or x ∈ B, or x ∈ C} = {x : x ∈ A or x ∈ B} ∪ C = (A ∪ B) ∪ C.

A similar argument proves that A ∩ (B ∩ C) = (A ∩ B) ∩ C. 

Theorem 1.2 (De Morgan’s Laws) Let A and B be sets. Then

1. (A ∪ B)′^ = A′^ ∩ B′;
2. (A ∩ B)′^ = A′^ ∪ B′.

Proof. (1) We must show that (A ∪ B)′^ ⊂ A′^ ∩ B′^ and (A ∪ B)′^ ⊃ A′^ ∩ B′. Let x ∈ (A ∪ B)′. Then x /∈ A ∪ B. So x is neither in A nor in B, by the definition of the union of sets. By the definition of the complement, x ∈ A′ and x ∈ B′. Therefore, x ∈ A′^ ∩ B′^ and we have (A ∪ B)′^ ⊂ A′^ ∩ B′.

1.2 SETS AND EQUIVALENCE RELATIONS 9

relation in which for each element a ∈ A there is a unique element b ∈ B such that (a, b) ∈ f ; another way of saying this is that for every element in

A, f assigns a unique element in B. We usually write f : A → B or A →f B. Instead of writing down ordered pairs (a, b) ∈ A × B, we write f (a) = b or f : a 7 → b. The set A is called the domain of f and

f (A) = {f (a) : a ∈ A} ⊂ B

is called the range or image of f. We can think of the elements in the function’s domain as input values and the elements in the function’s range as output values.

a

b c

a

b c

A B

A g B

f

Figure 1.1. Mappings

Example 4. Suppose A = { 1 , 2 , 3 } and B = {a, b, c}. In Figure 1.1 we define relations f and g from A to B. The relation f is a mapping, but g is not because 1 ∈ A is not assigned to a unique element in B; that is, g(1) = a and g(1) = b. 

Given a function f : A → B, it is often possible to write a list describing what the function does to each specific element in the domain. However, not all functions can be described in this manner. For example, the function f : R → R that sends each real number to its cube is a mapping that must be described by writing f (x) = x^3 or f : x 7 → x^3.

10 CHAPTER 1 PRELIMINARIES

Consider the relation f : Q → Z given by f (p/q) = p. We know that 1 /2 = 2/4, but is f (1/2) = 1 or 2? This relation cannot be a mapping because it is not well-defined. A relation is well-defined if each element in the domain is assigned to a unique element in the range. If f : A → B is a map and the image of f is B, i.e., f (A) = B, then f is said to be onto or surjective. A map is one-to-one or injective if a 1 6 = a 2 implies f (a 1 ) 6 = f (a 2 ). Equivalently, a function is one-to-one if f (a 1 ) = f (a 2 ) implies a 1 = a 2. A map that is both one-to-one and onto is called bijective.

Example 5. Let f : Z → Q be defined by f (n) = n/1. Then f is one-to-one but not onto. Define g : Q → Z by g(p/q) = p where p/q is a rational number expressed in its lowest terms with a positive denominator. The function g is onto but not one-to-one. 

Given two functions, we can construct a new function by using the range of the first function as the domain of the second function. Let f : A → B and g : B → C be mappings. Define a new map, the composition of f and g from A to C, by (g ◦ f )(x) = g(f (x)).

A B C 1 2

3

a

b c

X

Y

Z

f g

A C

X

Y

Z

g ◦ f

Figure 1.2. Composition of maps