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A friend of mine, who is a high school math teacher and majored in math in college, recently asked me for a good book to read on Abstract Algebra (presumably, group theory). She is looking for something to read semi-casually, so no serious textbooks which hide intuition.

My own education largely skipped the basics. I read through Pinter in the library one afternoon, and that was pretty much it. So, can people recommend a good book on abstract algebra/groups for the casual learner?

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Based on some of the answers here, it seems this question is confusing. Some have interpreted it as asking for a 2nd algebra course/book to read [casually]. So there are several suggestions for Galois theory books. On the other hand, unless the OP was speaking of himself in 3rd person, his friend probably had not read Pinter, so the question is [then] about a first book to read, not a second one. – Fizz Apr 12 '15 at 13:05

"A Book on Abstract Algebra" by Charles Pinter is a great book for the casual reader. It's an easy read yet maintains rigour throughout all of the topics discussed.

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The exercises in this book are especially worthwhile. – user12998 Nov 3 '11 at 0:08
Like I said in another thread on this topic, the main issue I see with Pinter's book is that too many essential algebra topics and proofs are relegated to exercises. Even graduate textbooks will contain brief proofs for some of those topics (isomorphism theorems, Sylow theorems, group actions etc.) Pinter's books instead spends an inordinate amount of time/space on stuff that you should know already, like what are functions or partitions of a set. – Fizz Apr 11 '15 at 10:44

A gentle introductory book full of intuition on group theory is Nathan Carter's Visual Group Theory. This answer is doubtless much too late for the OP, but may help others looking for casual books on abstract algebra.

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My favorite too!! – David G. Stork Apr 26 '15 at 3:05

She will love Fearless Symmetry by Ash / Gross. Mostly group theory but that's the best start, don't you think so?

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After a quick look I agree with the top-most Amazon review: The book is actually about Galois groups and how this theory lead to the recent (1995) proof of Fermat's last theorem. All of that is attempted "in a friendly style for a general audience", an ambitious undertaking to be sure. So not exactly what's being asked for here. Also he fist two chapters of the book talk a lot of geometry (rotations etc.) with nary a picture. – Fizz Apr 12 '15 at 12:57

Pinter is an excellent book, but I'd like to also recommend Abstract Algebra and Solution by Radicals, by John and Margaret Maxfield. It's very readable, and takes a semi-casual path through group theory, ending with elementary Galois theory and impossible constructions. This book was my first introduction to abstract algebra several years ago. If your friend wants a gentle introduction to the subject, I think it could be ideal.

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I second the recommendation of Maxfield/Maxfield. In fact, I was going to recommend it myself until I saw that you beat me to it. (By over two years, as I now happened to notice from your answer date!) – Dave L. Renfro Mar 20 '14 at 14:24

It depends the level of the casual reader (unless by casual you meant low prereqs). If you are looking for a book that requires a little more sophistication, but pays back two-fold in intuition I believe Artin's book Algebra is a good place to look. In particular, it helps motivate abstract groups by looking at the much more managable, understandable, notion of matrix groups.

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Field theory and its classical problems by Hadlock is wonderful.

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From the topmost Amazon review: This book gives a very concrete intro to Galois theory & field theory & would be an excellent supplement to an advanced course on algebra. Emphasis on "supplement" being mine. So that not what's being asked for here. – Fizz Apr 12 '15 at 13:00

It depends on what she wants out of the book, but it sounds like "Contemporary Abstract Algebra" by Gallian would be a great fit.

The book introduces all of the major concepts in a way that is both easy to grasp and motivated. There are also interesting historical insights placed throughout the book. I had a friend lend it to me when I was starting to learn abstract algebra and it definitely helped pique my interest.

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