Visual Group Theory (MAA Problem Book Series) 1st Edition

by Nathan Carter (Author)

34 ratings

ISBN-13: 978-0883857571

ISBN-10: 088385757X

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This text approaches the learning of group theory visually. It allows the student to see groups, experiment with groups and understand their significance. It brings groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory. Opening chapters anchor the reader's intuitions with puzzles and symmetrical objects, defining groups as collections of actions. This approach gives early access to Cayley diagrams, the visualization technique central to the book, due to its unique ability to make group structure visually evident. This book is ideal as a supplement for a first course in group theory or alternatively as recreational reading.

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Editorial Reviews

Review

Carter presents the grojp theory portion of abstract algebra in a way that allows student to actually see, using a multitute of examples and applications, the basic concepts of group theory...The numerous images (more than 300) are the heart of the text. As this work enables readers to see, experiment with, and understand the significance of groups, they will accumulate examples of groups and their properties that will serve them well in future endeavors in mathematics. Recommended --J. T. Zerger, Choice

If you teach abstract algebra, then this book should be a part of the resources you use. While the phrase "visual abstract algebra" may seem to be a contradiction, the diagrams in this book are an existence proof to the contrary. They are clear, colorful and concise very easy to understand and sure to aid the students that have difficulty in internalizing the abstract nature of the subject matter. Especially appealing are the colorized tables of groups and their operations.

The approach is a very slow one in the sense that a foundation of common operations and rearrangements that are groups that are first examined with text and images. A large number of exercises are included at the end of each chapter and detailed solutions with colored images found in an appendix.

this book could also serve as a text in a first course in abstract algebra provided that the course is limited to groups only or you used supplementary material for rings and fields. If your course is restricted to groups only, then this is the best book available. --Charles Ashbacher, Journal of Recreational Mathematics

Book Description

A full-colour visual guide for students of group theory.

Publisher ‏ : ‎ MATHEMATICAL ASSOCIATION OF AMERICA; 1st edition (May 1, 2009)
Language ‏ : ‎ English
Hardcover ‏ : ‎ 306 pages
ISBN-10 ‏ : ‎ 088385757X
ISBN-13 ‏ : ‎ 978-0883857571
Item Weight ‏ : ‎ 1.76 pounds

Best Sellers Rank: #1,046,845 in Books (See Top 100 in Books)
- #129 in Group Theory (Books)
- #1,307 in Algebra & Trigonometry
- #1,647 in Statistics (Books)

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34 ratings

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A. Menon

Very nice introduction to group theory- helps build the readers intuition

Reviewed in the United States on November 7, 2019

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Really nice introduction to groups and their applications in abstract algebra. Abstract algebra or analysis are usually a student's first introduction to having to write proofs and higher mathematics. They are typically very challenging as it requires training under a new regime, this book makes the transition relatively easy, and illustrates an often challenging subject with much ease. The book covers most material in a group theory course and ends with an overview of Galois Theory. It is accessible, digestible and illuminating look into abstract algebra for the beginner, though parts of it can be considered useful for those already familiar.

The book is split into 10 chapters starting with bringing up the concept of a group in the context of simple games with strict rules and reversible moves. From there the author highlights how such simple games constitute a group and how each of the moves is a group action and develops the idea of a group from simple intuitive phenomenon. The author then moves into techniques of visualization and introduces Cayley diagrams, he does it in simple forms that illustrate the essential ideas clearly to the reader. The approach of the author focuses at first on elements of the group representing actions rather than elements of a set, but explains the natural correspondence between the ideas. The author then gets into where groups come up and how they can be seen everywhere. The focus on symmetry properties is pronounced as finite groups or discrete groups representing symmetries have highly tangible visual representations in Cayley diagram form. The author then highlights the algebraic properties of groups and their consequences when looked at in multiplication table form. By clever use of coloring one can see how patterns can be found in groups via looking at the multiplication table. Such techniques are novel and give a quick deeper appreciation of the properties of a group. Such multiplication table graphics lead to a quicker understanding of things like subgroups and quotient groups. The author moves onto characterizing finite groups and effectively communicate properties of symmetric and alternating groups and present cayley diagrams in A5 which set the stage for Galois theory. The author tackles typical elementary topics like subgroups and cosets and illustrates key results like Lagrange's Theorem. The proofs are not terse, to some extent they are too conversational rather than straight to the point but for the uninitiated it makes the text very approachable. The author gets into other core topics like products and quotient groups and highlights the importance of normal subgroups for forming quotients. These ideas fall naturally into explaining homomorphisms, a central concept of group theory. The author then tackles some of the main undergraduate results of finite group theory, namely Sylow Theory. The author moves from Lagrange's theorem to Cauchy's theorem and then finally to Sylow's theorems. The author then spends a chapter on Galois theory which is light but illustrates the key idea of the Galois group of a polynomial. In particular the author weaves back in that A5 doesn't have a normal subgroup and so the quintic won't have a solution by radicals. Though this introduction to Galois theory is intuitive it does not cover the topic that thoroughly and leaves out material on symmetric polynomials for example.

Really nice relatively light introduction to abstract algebra. This isn't a great textbook as it misses a lot of key topics like Rings and Fields, but overall if one is looking for a different approach to algebra or some relatively light math reading, this is a really nice book which builds good intuiting. There are other undergraduate books which are much more complete but the novelty of the approach makes this a worthwhile addition to one's library.

3 people found this helpful

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Mark Saroufim

My favorite abstract math book

Reviewed in the United States on June 15, 2020

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There’s a mistaken assumption that algebraic proofs are more rigorous than visual ones. Just read Elements of Euclid by Byrne to convince yourself otherwise in the simple case of Euclidean Geometry. That said the Visual approach also applies to much more complicated mathematical structures and Visual Group Theory will give you beautiful diagrams which you can also play around with online.

One person found this helpful

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Amazon Customer

A great read for the mathematics hobbiest

Reviewed in the United States on December 24, 2014

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This is a great book for anyone interested in mathematics. I bought it just to read after reading about it in another text but find myself returning to it again and again to sharpen my understanding with the examples.

4 people found this helpful

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Alinak

Great book

Reviewed in the United States on August 30, 2019

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Easy to read and understand

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Laurent P

Great for develiping intuition and even for self-learning

Reviewed in the United States on November 3, 2014

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An outstanding introduction to the theory of groups. Great for develiping intuition and even for self-learning. An enjoyable book overall, very good editing with a nice visual appeal onevery page.

2 people found this helpful

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Thomas E.

Impressive book

Reviewed in the United States on May 22, 2019

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Met with universal praise within the house.

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Andrew

Great so far, through ~5/6 chapters

Reviewed in the United States on December 25, 2011

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I was a physics B.S. who is now pursuing a Ph.D. in applied physics. Was very intimidated by the very math-y and formal books on group theory. Have been working through all the problems through the first 5/6 chapters so far, the ones whose answers are in the back of the book. Already feel much more confident about the fundamentals of group theory. The referenced software the author created is also very helpful and creative. Plan to continue to work problems all the way to the end, after which I'm confident I'll feel like I've got a solid grasp on group theory.

11 people found this helpful

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I really wanted to love this book but...

Reviewed in the United States on May 7, 2010

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...I only like it. Given that I am not a fan of the formal turgid theorem/proof type of math text, I waited with much
anticipation to get a copy of this text. Without a doubt, this is a very original and fun text that presents a novel
approach to teaching group theory. However, if I had come to group theory with this text as my first
introduction, I do not think it would have been effective and I would have quickly become frustrated. Simply put, the author
goes to far to build up intuition about groups before he actually defines a group mathematically.
I think it would have been better to define a group early and then start to build up
intuition after the definition is secure. Sort of "Here is the abstract definition, lets now start to understand why anyone would want to define a group in the first place and lets see some examples involving symmetry and other physical situations where groups arise".

A better choice for the absolute novice in group theory is the book Groups by Jordan and Jordan. This book is not well known in the USA but is simple, intuitive, and
well-written.
For the person interested in exploring the entire landscape of abstract algebra (groups, rings, fields, Galois theory)
"A Book of Abstract Algebra" by Pinter is unmatched for its clarity.

In summary, I cannot recommend Visual Group Theory as a textbook for a course in group theory although some might find it a refreshing supplement to an overly formal abstract algebra text/course. I think its target audience
falls outside workers in the mathematical sciences and is geared toward chemists, molecular biologists, future/current
high school math teachers, and weekend math warriors
who are looking for intellectual stimulation.

77 people found this helpful

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Top reviews from other countries

Amazon Customer

fantastic

Reviewed in the United Kingdom on April 19, 2011

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Excellent! After searching so long for a good book that takes you from zero to significant understanding but in an intelligent way (visual here) I've found a book that does it.

Writing is clear. Exercises are very well thought out and not there just to add volume to the book. You should do them.

So much of mathematical/scientific writing is almost deliberately obscured by jargon or bad writing. In fact much of the material can be expressed to most intelligent or curious readers .. if it's presented and written well.

Great book - and unlike some books this is printed very well on good materials.

If I could have more I'd ask for more material on the symmetries in number theory.

7 people found this helpful

Robert Scarth

Wonderful book

Reviewed in the United Kingdom on February 16, 2013

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I really enjoyed reading this book. It is an introduction to group theory, and could be read and enjoyed by anyone with minimal background in mathematics. I studied Group Theory for my PhD (although that was more than ten years ago now) but even so, I still learnt things from this book. The tool most used in the book are Cayley diagrams, and the diagrams in the book are wonderfully clear. The goal is to explain the basic concepts of group theory, and to build up the reader's intuition about groups - essential in what can often be a very dry and abstract subject. The book achieves this goal brilliantly, and several ideas and results become essentially obvious. The introduction of semi-direct products is the best I have ever read, and why the fundamental theorem of abelian groups is true is made so obvious it is almost trivial.

The final two chapters cover Sylow's Theorems and Galois Theory. The Sylow Theorems are clearly proved and groups up to order 15 are classified. The final chapter gives a brief introduction to Galois Theory. Little is proved in this chapter, but the reader is given a sketch of what the theory is about and roughly how it works. It is an excellent taster for further study.

The book contains a multitude of exercises which are mostly fairly straight forward, and if I had to criticise the book it would be for the lack of stretching exercises, but this is a very minor criticism; there are many other books which can stretch a keen student of group theory.

This is an excellent book, and would be perfect reading for anyone beginning their study of group theory.

6 people found this helpful

phorth

Enables an intuitive grasp of Group Theory

Reviewed in the United Kingdom on November 24, 2016

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I managed to pass the Group Theory module at the Open University but still hadn't got an intuitive feel for what cosets/ normal groups etc. were. This book really helps you get a gut feeling as to what the various aspects of group theory are and how they are useful.
One point to readers: even if you think you know group theory, it is well worth (essential?) to got through the exercises.

One person found this helpful

pw242

Finally understand group theory

Reviewed in the United Kingdom on August 2, 2013

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I struggled with groups at university. Although I was able to stumble through questions and manipulate the algebra I didn't have any intuitive feel for what a group really 'was'. This book succeeds in providing that intuition.

5 people found this helpful

Davide Berti - Cuneo

Mathematics for human beings

Reviewed in France on February 15, 2014

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I think this book can be enjoyed by anyone. Starting from the Rubik cube it sets his way to theory groups up to Sylow theorems and elements of Galois theory on algebraic equations, among really beautiful and useful diagrams and explanations very easy to follow but not at all stupid. I recommend this especially to teenagers for inspiration on mathematics, and to their teachers too.