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Visual Group Theory
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.
306 pages, Hardcover
First published June 1, 2009
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Displaying 1 - 9 of 9 reviews
September 17, 2020
This is a good introduction to group theory for beginners. It starts off slowly and informally enough to be accessible to someone with no abstract algebra background, and uses diagrams very effectively. You will have a better time if you have physical models of the platonic solids to consult. There are lots of nice exercises, many of which are pretty gentle.
April 4, 2013
it's a 5-star idea, group theory certainly calls for visual treatment, I hold back a star only because it still feels like a first draft, alternately too verbose and too terse, didn't find it as useful as I hoped as a complement to more opaque texts, seemed a little fluffy still.
May 11, 2021
Beautiful introduction to group theory requiring almost no background.
I was astounded by this book. It explains group theory in a way that makes me feel like I've got a really solid understanding. It requires only school level maths, yet takes you into Sylow and Galois theory in a way that feels incredibly non-trivial.
I was reading it to brush on group theory, as I've recently been using it in my research, and I absolutely loved it. I thoroughly recommend to anyone interested in maths who wants to get an alternative, visually oriented, introduction to group theory.
I was astounded by this book. It explains group theory in a way that makes me feel like I've got a really solid understanding. It requires only school level maths, yet takes you into Sylow and Galois theory in a way that feels incredibly non-trivial.
I was reading it to brush on group theory, as I've recently been using it in my research, and I absolutely loved it. I thoroughly recommend to anyone interested in maths who wants to get an alternative, visually oriented, introduction to group theory.
February 15, 2021
Visual explanation works best for me. The best book to explain Group theory in an intuitive way. Video lectures and supplementary tools (Group Explorer) are very useful.
Évariste Galois was a rockstar )
Évariste Galois was a rockstar )
July 30, 2019
My views on this book are certainly heavily influenced by the fact that I already knew almost everything that appears in it, but I will still give it an honest go.
It starts of slow, probably too much so, but the start managed to change my thinking of group theory from algebraic to the visual one, at least for the purposes of following the book.
Still, most of the things expressed visually just don't seem like the better or more intuitive way to go. Rather, they are nice ways to visualize things that are easily understood from a strictly algebraic point of view.
There are only two things in the book that I found useful to visualize were semi-direct products and the idea of regularity of Cayley diagrams, in other words the local structure which subgroups impose on the whole group. I'm not sure how useful they will be in my future study of groups, but they are interesting on their own.
The later parts of the book stop being heavily visual, but that turns out to be a good thing.
By far the best part of the book were the highly systematic and intuitive proofs of Sylow theorems.
Parts of the proof are given visualization, but here it becomes obvious that the intuition and motivation for the proofs stops being visual.
The proof of non-solvability of quintic polynomials was also great. While skipping almost all proofs, the entire exposition is intuitive and convincing, and again, not visual.
I the end, I think that only small parts of the visual idea are useful, and although it is a cool alternate point of view, so I am not sure that it warrants writing an entire book.
Keeping that in mind, and the fact that some parts of the book are amazing, 4 out 5 stars.
It starts of slow, probably too much so, but the start managed to change my thinking of group theory from algebraic to the visual one, at least for the purposes of following the book.
Still, most of the things expressed visually just don't seem like the better or more intuitive way to go. Rather, they are nice ways to visualize things that are easily understood from a strictly algebraic point of view.
There are only two things in the book that I found useful to visualize were semi-direct products and the idea of regularity of Cayley diagrams, in other words the local structure which subgroups impose on the whole group. I'm not sure how useful they will be in my future study of groups, but they are interesting on their own.
The later parts of the book stop being heavily visual, but that turns out to be a good thing.
By far the best part of the book were the highly systematic and intuitive proofs of Sylow theorems.
Parts of the proof are given visualization, but here it becomes obvious that the intuition and motivation for the proofs stops being visual.
The proof of non-solvability of quintic polynomials was also great. While skipping almost all proofs, the entire exposition is intuitive and convincing, and again, not visual.
I the end, I think that only small parts of the visual idea are useful, and although it is a cool alternate point of view, so I am not sure that it warrants writing an entire book.
Keeping that in mind, and the fact that some parts of the book are amazing, 4 out 5 stars.
January 3, 2013
"By now the reader is certainly convinced that group theory shows up in diverse situations. But it would be a great disservice to the history of mathematics if I did not mention one more application, the very reason that group theory was invented! In the nineteenth century, two young mathematical prodigies, Neils Abel and Evariste Galois, solved a mathematical problem that had stood unsolved for centuries. It has come to be called 'the unsolvability of the quintic.' It is one of the great discoveries in mathematics, and when you come to Chapter 10 of this book, you will be ready to read about it in some detail. It rests on the fact that the solutions to polynomial equations have a certain relationship to one another. They form a group."
May 19, 2014
Delightful introduction to group theory. Heavily designed around using pictures [Cayley diagrams, Hasse diagrams, multiplication tables] to build intuitions. The presentation is pitched somewhat lower than a first serious abstract algebra course; many of the late results [Sylow and Galois theory] are done as proof sketches rather than proofs, with many unproved claims. May be suitable for advanced high school students.
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November 26, 2013Well, we're breaking even on the Acknowledgments page, with a +1 for the Hofstadter lineage (I think he was the first, or among the first, to teach a visual group theory course a number of years ago) and a -1 for thanking "God for life and breath and mathematics, as well as my ability to write it, draw it, and enjoy it."
May 11, 2015
Lots of intuition on groups.
You should read this as a supplementary text.
You should read this as a supplementary text.
Displaying 1 - 9 of 9 reviews