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I'm familiar with rigorous linear algebra, and I've had a very elementary course on modern algebra. I'm not interested in algebra but I need to learn more about it. Hence I'm looking for a concise and self-contained book on abstract algebra which covers what is needed for applications in the parts of mathematics relevant to physics, esp. differential geometry (incl. Lie theory and de Rham cohomology) and operator algebras.

I'm not sure what topics exactly the book needs to cover, but probably someone here does. I'm also not sure if such a book exists: perhaps these areas are too broad. It doesn't have to literally be a book: it could be a chapter or appendix in another book, or lecture notes, but it should include nontrivial proofs. It would be best if the book assumes a knowledge of linear algebra so that the general linear group, etc. can be used as examples.

To clarify: of course I'll have to look up specialized topics in one of the encyclopedic books, but I'm trying to find something that quickly covers the basics.

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    $\begingroup$ A word of caution: in math, short rarely coincides with fast. For example, a book like Dummit and Foote, while enormous, is relatively easy to fly through if you're willing to ignore some examples, harder proofs, etc. Something like Lang's Algebra is terse but includes an absurd amount of information, and is known for being quite difficult to get through. Try Michael Artin's "Algebra". It's reasonably quick, doesn't have too many advanced/specialized topics, and is very rooted in linear algebra. $\endgroup$ – Dorebell Jun 24 '15 at 22:02
  • $\begingroup$ I would like to add to the previous comment that nevertheless Lang's proofs of Sylow Theorems are one of my favorites. $\endgroup$ – Pedro Sánchez Terraf Jun 24 '15 at 22:22
  • $\begingroup$ @Dorebell I'll agree with the examples in Dummit and Foote. Some of those examples are flat out brutal to read through ! $\endgroup$ – TuoTuo Jun 24 '15 at 22:41
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Elements of Abstract Algebra by Allan Clark is the shortest book I've ever seen. This book is a little strange in that it covers Field and Galois Theory before ring theory if I remember correctly. It's also not a "hand holding" book and it expects you to do some work to read through it. This can be good or bad depending on the individual.

Serge Lang's books are also usually pretty short but I've never liked his books.

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I don't have experience with this personally, but colleagues of mine who do differential geometry said that they enjoyed "Naive Lie Theory" by Stillwell. The reviews on Amazon are also very favorable.

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  • $\begingroup$ Thanks for the input, but I'm familiar with that book and it's not what I'm looking for. $\endgroup$ – Randy Randerson Jun 24 '15 at 21:55
  • $\begingroup$ I see. Well, too bad. I hope someone else has a better suggestion then. $\endgroup$ – Joel Jun 24 '15 at 21:56

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