Tractability and the Fine Art of Mathematical Groups: Teach Yourself Mathematical Groups
Written: Sep 12 '03 (Updated Sep 13 '03)
Product Rating:
Pros: Short, concise introduction to Group Theory and Modern Algebra
Cons: Not too insightful, can be too abrupt at times.
The Bottom Line: The best introduction to Group Theory. A cheap means of gaining a firm grasp of Algebraic structure. A must for early college math students. Lacks deep insight though.
avepythagoras's Full Review: Mathematical Groups Books
On the topic of math:
Math and I have always been a dysfunctional couple. I never really bring her to parties, and always forget our anniversary. And she is a harsh mistress, demanding my utmost attention and fidelity. Sometimes we get in long, brutal arguments and I end up needing some space. We part ways and I date other, less involved, subjects. Mostly out of spite, but generally because I get lonely at nights and need the comfort of a warm textbook. But I always find myself later, on hands and knees, begging forgiveness from my love. We always end up back together, to start this codependent cycle anew. I could never marry her, but by god, I don't know what I'd do without her.
Oh wait this is a book review:
Math. She is an elusive, crazy subject. Enigmatic and hard to please. I need all the help I can get. So occasionally I look beyond the standard textbook fare for little cheap novelties able to help me comprehend the further depths of her rigorous being. Teach Yourself Mathematical Groups is one such example: a short, cheap, paperback overview of the finer mechanics of Modern Algebra and Group Theory. But this book packs a particular punch. You can actually learn quite a bit from it. And would suggest using it as a primer for more advanced mathematical instruction.
These 'course overview' books generally aren't worth anyone's time. Most overviews are useful only after you've taken the course, have mastered the key functional elements and find yourself, years down the line, in need of a brief refresher. Mathematical Groups is more than a refresher; it is intended to provide a self-paced introduction as well as an easy reference for later "jump starts" years down the line. In short, it is a book that will keep you fresh, regardless of your familiarity of the subject, for the most part.
As An Introduction: Mathematical Groups
Math isn't concerned as much with numbers as it is with sets, or objects within sets. We are generally familiar with numbers and have studied them at some point in our lives. We know how to add, subtract, multiply, divide, but what many fail to realize is these mathematical operations are only isolated instances of a greater, or more 'abstract', system of mathematics. In fact, mathematical operations are, for the most part, arbitrarily defined. We can make them up as we go and make them do things other than add, subtract, or multiply. Modern Algebra is the study of these operations and the effect they have on sets. What properties will these sets, read numbers, display if we define them under different operations? Will the rules of arithmetic still hold? Or will we have an entirely new system of algebra? Will it be more complex or easier to manage? Can we compare this new system with the familiar number system of arithmetic? Such questions are addressed and answered by Modern Algebra, a rigorous system theories describing the behavior of sets under specifically defined rules. A group, for instance, is one just particular 'behavior' numbers can take.
Mathematical Groups provides a reasonable introduction to the theory of Groups and provides the basic understanding needed to pursue further study in Modern Algebra. As an introduction to the subject, it is very rudimentary and easy to follow, though it does require a certain level of mathematical insight. Equivalent, perhaps, to the level of rigor encountered in an entry-level Calculus course. The book is fairly successful, if not a bit terse, in presenting key ideas and theorems. Most of the important foundational concepts are studied and emphasized with concise examples. It presents proofs quickly and efficiently, but on occasion moves a little too fast. Each chapter ends with a short body of questions, all of which are answered within the appendix. This is an ideal book for self-study. Nothing is left to the imagination. The writers keep the goal of the student firmly in mind and try to make this book as accessible as possible. Even so, the later chapters grow a little cloudier, the logic more difficult, making the book hard to follow at times. But, because this book is self-contained, all of the information required to understand the proofs has been discussed, and one needs merely flip back a few pages to refresh.
For what it is worth, about $13 dollars, I'd say this is one of the best introductions to higher mathematics: both cost and educationally efficient. But because of its brevity and function, namely as an introduction, it packs little, if any, insight into the deeper mechanisms of Algebra. The information is cursory, a good stepping stone but by no means complete in-itself. Mathematical Groups is an exceptional book for aspiring mathematicians, a good companion or primer for those desiring a better foundation in the 'numerical' arts. I would even suggest it to accelerated high school students who want to get the jump on a future career in theoretical mathematics--hey, it would've helped me. I can't really recommend this book to practicing mathematicians other than as a refresher, of which, at that point, there are better sources. For further study, try Gallian's masterpiece Contemporary Abstract Algebra. This book is one of the best of its kind. I would recommend Mathematical Groups to anyone curious about higher mathematics, or those who are just simply curious, bored with the common human condition, desiring something more than just 'the obvious'.
In the end:
Ah, the joys of my love with mathematics. I don't think I'll ever completely understand her, and I'm sure we're destined to some choice arguments in the future. But she is the love of my life. And I cannot see myself with any other woman...er subject.
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