
Elementary Abstract Algebra: Examples and Applications
Justin Hill, Temple College
Chris Thron, Texas A&M University-Central Texas
Copyright Year:
ISBN 13: 9780359042111
Publisher: Justin Hill and Chris Thron
Language: English
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Conditions of Use
Attribution-NonCommercial-ShareAlike
CC BY-NC-SA
Reviews





Extensive index is included. The book includes more topics than a semester course could study. read more
Reviewed by Mark Koester, Associate Professor, Metropolitan State University of Denver on 3/26/21, updated 5/26/21
Comprehensiveness
Extensive index is included. The book includes more topics than a semester course could study.
Content Accuracy
There appears to be accuracy in the book.
Relevance/Longevity
These topics in Abstract Algebra are appropriate for secondary preservice mathematics teachers. Many of our state's mathematics standards connect to the topics in the book.
Clarity
I enjoy the conversational style of the authors. It is so important that textbooks are accessible for all.
Consistency
The same terminology is used throughout.
Modularity
There are twenty chapters. Of course, some ideas build on others, but often, it is noted in what chapter these other ideas can be found.
Organization/Structure/Flow
There is a logical order to the text.
Interface
I like having the ability to click on hints or propositions n the book rather than being inundated with text on a page.
Grammatical Errors
I didn't notice grammatical errors.
Cultural Relevance
I believe that this book is inclusive of others because it is so readable. There is not the inherent bias that one must be able to read cryptic symbols or to read the author's words obviously, clearly, or this is easy. This was so damaging to having a positive identity as a mathematics major.





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
Reviewed by Diana Morris, Assistant Professor, University of Virginia on 12/13/20
Comprehensiveness
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.
Content Accuracy
Given the trade-off between perfect mathematical clarity (think Rudin) and being intelligible to undergraduates, I think the book strikes a good balance.
Relevance/Longevity
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?)
Clarity
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.
Consistency
The material has numerous different sources. None of the transitions were jarring, but sometimes it felt more like a compendium than one flowing volume.
Modularity
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.
Organization/Structure/Flow
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.
Interface
The only thing that made it difficult to navigate was the sheer length of the document.
Grammatical Errors
The book had its share of misspellings, but not to the point of compromising or distracting from the meaning.
Cultural Relevance
The book did an excellent job of relating abstract mathematical concepts to practical examples familiar to any person.
CommentsI 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
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
Ancillaries
- Justin Hill and Chris Thron (by request)