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.
Best Answer
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.