# Elementary Abstract Algebra: Examples and Applications

Justin Hill, Temple College

Chris Thron, Texas A&M University-Central Texas

Copyright Year: 2019

ISBN 13: 9780359042111

Publisher: Justin Hill and Chris Thron

Language: English

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

## Reviews

At almost 1000 pages, this book certainly covers the basics comprehensively. You might not find the Sylow Theorems or Jordan Canonical Form, but it accomplishes what it sets out to do, covering the basic background of numbers and set theory, the... read more

At almost 1000 pages, this book certainly covers the basics comprehensively. You might not find the Sylow Theorems or Jordan Canonical Form, but it accomplishes what it sets out to do, covering the basic background of numbers and set theory, the main examples of groups and rings, and then developing these concepts more formally.

Given the trade-off between perfect mathematical clarity (think Rudin) and being intelligible to undergraduates, I think the book strikes a good balance.

Abstract algebra books typically struggle to convince undergraduates of their relevance. The sustained application in the book is Cryptography, but they did a beautiful job of relating basic concepts like modular arithmetic to practical applications that we do every day (e.g. if December 1st is on a Tuesday, what day of the week is Christmas?)

Their basic wording of mathematical concepts was impressive, e.g. the explanation of multiplicative and additive identities is completely fool-proof, as were the every day examples of mathematical concepts, e.g. the notion of "grandfather" being a composition of "parent" and "father". There is a real effort to guide students through proofs, e.g. how to justify each step and what qualifies as justification. I also appreciated the generous hints on the exercises spread throughout the book and the Study Guides at the end of each chapter identifying competencies.

The material has numerous different sources. None of the transitions were jarring, but sometimes it felt more like a compendium than one flowing volume.

Given the length of the text, an instructor would certainly want to pull out different chapters and make particular note of which leads to which. There is actually a chart at the beginning of the text explaining how sections are related and which rely on previous sections.

I could not decide if the organization of the book was confusing or brilliant. For example, major examples of groups / rings are first defined in great depth without formally defining them as such, then in later chapters, one starts from the beginning with the definitions of groups and rings and discovers that they are already familiar. I did feel that there was an overwhelming amount of material, more a reference book of basic abstract algebra topics than a simple narrative, but there were highly coherent sub-narratives throughout.

The only thing that made it difficult to navigate was the sheer length of the document.

The book had its share of misspellings, but not to the point of compromising or distracting from the meaning.

The book did an excellent job of relating abstract mathematical concepts to practical examples familiar to any person.

I would certainly recommend this book as a reference to any student wanting simple explanations and practical examples for the concepts of basic abstract algebra. For most single semester introductory courses, it would make a fine basic textbook, with some selective pruning and perhaps a few additions where the instructor sees fit.

## Table of Contents

- 1 Preliminaries
- 2 Complex Numbers
- 3 Modular Arithmetic
- 4 Modular Arithmetic, Decimals, and Divisibility
- 5 Set Theory
- 6 Functions: Basic Concepts
- 7 Introduction to Cryptography
- 8 Sigma Notation
- 9 Polynomials
- 10 Symmetries of Plane Figures
- 11 Permutations
- 12 Introduction to Groups
- 13 Further Topics in Cryptography
- 14 Equivalence Relations and Equivalence Classes
- 15 Cosets and Quotient Groups (a.k.a. Factor Groups)
- 16 Error-Detecting and Correcting Codes
- 17 Isomorphisms of Groups
- 18 Homomorphisms of Groups
- 19 Group Actions
- 20 Introduction to Rings and Fields

## Ancillary Material

- Ancillary materials are available by contacting the author or publisher.

## About the Book

This book is not intended for budding mathematicians. It was created for a math program in which most of the students in upper-level math classes are planning to become secondary school teachers. For such students, conventional abstract algebra texts are practically incomprehensible, both in style and in content. Faced with this situation, we decided to create a book that our students could actually read for themselves. In this way we have been able to dedicate class time to problem-solving and personal interaction rather than rehashing the same material in lecture format.

## About the Contributors

### Editors

**Justin Hill**, Temple College

**Chris Thron**, Texas A&M University-Central Texas