Visual Group Theory, Volume 32 (Classroom Resource Materials)

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.

I've had lots of math but group theory is a bit different. It uses some of the same terms but their meaning can differ and some are completely different. When I got this book, it was an attempt to understand the use of group theory in particle physics. Upon studying the subject, I decided that group theory needed to be studied as a stand-alone subject and then after that, see about how it applies. Nathan Carter's approach to group theory using visual demonstrations as he goes along and adding in the more traditional symbolic or as it is called algebraic method worked to give me a way into understanding this subject. This approach was diminished as I got to the last part of the book. The subjects became more complex so didn't lend themselves as much to a visual approach but not entirely. All-in-all the only negative for this book is that answers needed to be provided for all the exercises. And not just answers, how the problems were worked would greatly help. I tried to work them but not knowing if I got them correct was less than helpful. YouTube has a video series that parallels this book presented by Professor Matthew Macauley. His course varies a bit but they complement each other quite well. I'm far from an expert on the subject now, but I can at least understand a significant portion of its use. It has served well as a great starting point.

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.

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.

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.

I am a self study student of mathematics having acquired a taste for it later in life. I only had 1st year calculus and linear algebra in university nearly 20 years ago. This book is a very readable introduction to group theory. I suppose it lacks some of the rigor a truly dedicated mathematician might require, but I have really been enjoying teaching myself group theory from this book. There are lots of examples to think about and many problems to work through. Very readable.

Met with universal praise within the house.

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.