Visual Group Theory (MAA Problem Book Series) 1st Edition
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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
- 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: #403,863 in Books (See Top 100 in Books)
- Customer Reviews:
Top reviews from the United States
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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.
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
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.
Top reviews from other countries
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.
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.
One point to readers: even if you think you know group theory, it is well worth (essential?) to got through the exercises.