Abstract Algebra

Introduction to Groups, Rings and Fields with Applications

Clive Reis, Stuart A Rankin;;;

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789814730563
• Edition: 2
• Topic: Mathematics
• Subtopic: Abstract Algebra
• Index: Mathematics

This second edition covers essentially the same topics as the first. However, the presentation of the material has been extensively revised and improved. In addition, there are two new chapters, one dealing with the fundamental theorem of finitely generated abelian groups and the other a brief introduction to semigroup theory and automata.

This book is appropriate for second to fourth year undergraduates. In addition to the material traditionally taught at this level, the book contains several applications: Polya–Burnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes, and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space, semigroups and automata. It is hoped that these applications will help the reader achieve a better grasp of the rather abstract ideas presented and convince him/her that pure mathematics, in addition to having an austere beauty of its own, can be applied to solving practical problems.

Considerable emphasis is placed on the algebraic system consisting of the congruence classes mod n under the usual operations of addition and multiplication. The reader is thus introduced — via congruence classes — to the idea of cosets and factor groups. This enables the transition to cosets and factor objects to be relatively painless.

In this book, cosets, factor objects and homomorphisms are introduced early on so that the reader has at his/her disposal the tools required to give elegant proofs of the fundamental theorems. Moreover, homomorphisms play such a prominent role in algebra that they are used in this text wherever possible.

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Chapter 1

Logic and Proofs

1.1 Introduction

Mathematics in the ancient world was born of practical necessity. In ancient Egypt, the yearly flooding of the Nile obliterated boundaries between properties which meant that these boundaries had to be redrawn once the waters receded, thus giving rise to a system of linear and areal measurement. In other ancient civilizations of the Middle East, increasing trade brought about the evolution of the concept of number to keep a tally of goods – by volume or by weight – and of earnings.

Out of these practical needs there grew a considerable body of “facts” and “formulae”, some valid and some merely poor approximations. It was not, however, until the classical Greek period (ca 600-300 BC) that mathematics emerged as a deductive discipline wherein the validity of statements and formulae required justification. It was no longer acceptable to assert that a 3-4-5-triangle is right angled (a fact that was known to the ancient Egyptians); a “proof” was required.

What precisely is a mathematical proof of an assertion? It is a finite sequence of propositions, each deduced from previously established truths and ending with the required assertion. A moment’s reflection will lead us to the conclusion that we must ultimately base our sequence of propositions on some fundamental assumptions which we agree to accept without proof; otherwise, we would be involved in an infinite regress by proving proposition A from B, B from C, C from D, etc.

These fundamental assumptions are called axioms or postulates and serve as a foundation upon which to build the various branches of mathematics. Later on we shall be studying a number of mathematical systems, among them groups and rings, and we shall see that each has its own set of axioms from which we shall deduce the various properties of the respective systems.

Since proofs are at the very heart of modern mathematics, it is essential that we familiarize ourselves with the fundamentals of deductive reasoning. This chapter is therefore devoted to making the reader aware of some of the simple rules of logic.

1.2 Statements, Connectives and Truth Tables

1.2.1 Definition. A statement is an assertion which is either true or false. “True” and “False” are the truth values of the statement, and these are abbreviated as T and F, respectively.

1.2.2 Example.

(i) The sea is salty. (Truth value T)

(ii) The moon is made of blue cheese. (Truth value F)

(iii) 4 is an even number. (Truth value T)

(iv) This is a beautiful painting. (Since beauty is subjective and different people may differ in their assessment of the truth value of this assertion, we deem this assertion not to be a statement.)

We denote statements by lower case letters. For example we might let p stand for the statement “The sea is salty”. We would indicate this by writing: p: the sea is salty.

We string two or more statements together to form a compound statement (or statement expression using logical connectives. The truth value of the compound statement naturally depends on the truth values of the components comprising the compound statement. We exhibit this dependence by means of a truth table as shown below.

The connectives we shall be using are: ¬, ∧, ∨, →, and ↔.

The Connective ¬ (negation).

This connective is read “not”.

1.2.3 Example. ¬ is read “not ”. Hence if p stands for “the sea is salty”, then ¬p stands for “not (the sea is salty)”, which in English we would express as “the sea is not salty”. If this connective is to ...