Most Helpful Customer Reviews
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65 of 71 people found the following review helpful:
5.0 out of 5 stars
A pleasure for all who see the world in terms of math, January 5, 2003
This book is more than a find --- it's a treasure chest. Gallian does more for the reader than just throwing a bunch of theory at them and laughing as the reader struggles to understand proofs and apply them himself; he provides a multitude of examples. And by a multitude, I don't mean three or four comprehensive examples per chapter: I mean three or four *per proof*. And often Gallian steps back a moment to discuss the proof or theorem he just outlaid in more conceptual terms, sometimes just as good as an example. The book is a delight, and absolutely perfect for the independent learner. Gallian gives a good background on integers, modular arithmetic, induction and the like in his chapter 0, and then eases his way into group theory in the next two chapters. He goes on to then concentrate on the most important forms of group the student will use, taking him through subgroups, cyclic groups, and permutation groups. Then he outlines some of the most important tools the reader will use when he's trying to deduce the properties and structure of a groups, like isomorphisms, cosets and Lagrange's Theorem, external and internal direct products, normal subgroups and factors groups, and homomorphisms. In the third part of the book, Gallian teaches rings. This section is set up much like the first section; he goes over the structure of rings themselves and then elaborates on tools which will help the mathematician to deduce structure and properties of rings more easily. Part four of the book, on fields, is set up much the same way. Part five of the book is on special topics: sylow theorems, simple groups, symmetry groups, crystollagraphics groups, galois theory, etc.. The most important feature of the book overall is its plethora of exercises and answers. Gallian averages about 45 exercises *per chapter,* and every four chapters or so there is another set of supplementary exercises covering the previous material. And there is a wonderful appendix in the back of the book which has the answers (some more detailed than others) to every other problem. In conclusion, this book is perfect for the independent learner, and a dream for a professor who seeks a textbook of ultimate clarity. This is the best textbook on advanced mathematics I have ever come across, accessible to a (bright) freshman in college.
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26 of 28 people found the following review helpful:
5.0 out of 5 stars
Real joy to read!, June 19, 2000
I had to learn group theory in a hurry for my research, and a friend recommended this book to me. Abstract algebra can very quickly become boring, but Gallian's writing makes it all fun.Gallian is clearly very passionate about the subject, and covers the basics of abstract algebra very well. The presentation in this book is fairly verbose, and therefore does not cover very much of the more advanced topics in abstract algebra. Nonetheless, after reading this book I feel sufficiently confident and enlightened to continue my journey into abstract algebra. This book is highly recommended for beginners interested in starting learning abstract algebra.
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13 of 13 people found the following review helpful:
5.0 out of 5 stars
Truly a gem of mathematical literature, December 21, 2005
I knew I was going to at least smile when I read this book because of the fact that Gallian uses Beatles' song lyrics to open some of his chapters. But, I was concerned about the mathematical content. Those fears were put to rest basically the minute I turned the first page.
His clarity in stating and proving theorems as well as providing NUMEROUS concrete examples (over 20 in the introductory group chapter alone!) at every point in your algebraic journey make this book a terrific read. He's also willing to go back and review a previous theorem or example if it illustrates a point; another excellent feature. Not only that, but his problems are well-chosen; not only are there so many of them, but there are "warm-up" problems to get your hands dirty right away and test your understanding of the basics, followed by some harder problems and some very interesting (i.e., difficult!) problems in each section. He also includes supplementary problems to get you to extend your understanding with new groups, definitions, etc. Whenever I get confused about anything in algebra, I can usually turn to Gallian to help me out.
Some have complained about the emphasis on group theory as opposed to ring/field theory. I don't see this as a hindrance, but rather as an advantage because, in a first algebra course, I feel learning the ins and outs of group theory as detailed as possible makes one really ready to study rings, fields, modules, etc. and then go back and say, "Hey! This isn't so hard; it's just like (blah) with groups!" You also get the advantage of having selected answers and hints in the back to check your work along the way as well as more advanced topics like the beginnings of Galois Theory, crystal and frieze groups, symmetry groups, etc. The book, regrettably, is missing a detailed treatment on group actions (though you do use them throughout, just not calling them that!) and some advanced linear algebra, but this is not enough for me to downgrade the book! For a book its size, it packs a lot of information and, in fact, I can't think of a single grad student in my department who either doesn't have a copy of the book or has at least looked at it when doing homework; it is really THAT good! Thanks, Gallian!
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