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I am searching for a book on group theory that follows in the style of textbooks written for math students (since I was one), but that covers all (or at least most) topics that would be needed in physics, such as in QFT for example (Think of the topics in Zee's Group theory in a nutshell).

My problem with Zee is that it is too verbose and not as mathematically rigorous as I'd prefer (I can understand the concepts better when they are presented in a formal mathematical way). And my problem with other standard group theory books for math is that they usually do not cover all the topics needed for physics (Lie Algebras, Poincare groups...)

A single or many-book-combination recommendation that would fill in these criteria is welcome.

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I was recommended Brian C. Hall's Lie Groups, Lie Algebras, and Representations for this purpose. It is distinct from standard textbooks on Lie Theory in that it assumes very basic mathematical preliminaries (basic definitions from finite group theory, topology, and linear algebra) and deals only with Matrix Lie Groups (i.e., is very concrete). Matrix Lie Groups are some of the most important Lie Groups considered in a physics context anyway. Even though the book presents Lie Theory more concretely, it is a graduate-level math textbook, and you get the same mathematical rigor you expect.

In particular, because the book is so concrete I was recommended this book as a first foray into Lie Theory (which it sounds like is what you are asking about). A more abstract Lie Theory text could follow after reading through this one.

It seems like you may be familiar with Differential Geometry already. In that case, I think you can skip straight to a more abstract Lie Theory textbook, but I unfortunately do not have any recommendations for such a book.

Other "standard group theory books" perhaps cover finite group theory (something you'd work through in a first course in Abstract Algebra), which to my understanding is not so useful for physics. Looking for books on Representation Theory or Lie Theory may come up with textbooks more suitable to your needs.

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I like "Lie algebras in particle physics" by Howard Georgi link. It is written like a math book - with all the required rigor for properly understanding the subject. It starts with discrete groups (and their uses in classical mechanics), then goes to Lie groups, all the group related concepts needed in the standard model, and finally all the grand unified theories with bigger symmetries. It also has homework questions, which I like.

And I agree that Zee is frustrating for someone who likes mathematical rigor. There's hand waving, and sometimes I feel you're left with a psudo-understanding.

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It is old, having first been published in the 60s, but I quite like Hamermesh's "Group theory and its applications to physical problems" (disclaimer: I have only read the first three chapters, and selected pieces of the rest). The other answers have already mentioned more modern books, so I add this for completeness.

It is more purely mathematical than many other books on the subject. The first two chapters introduce the basic notions of a group. Linear vector spaces, representations, and Hilbert spaces are introduced in chapter 3.

The type of group theory you need in QFT (i.e. Lie groups and the representation theory thereof) is introduced in chapters 8 ('Continuous groups'), which is already halfway through the book.

Despite the title, physical applications are a bit thin on the ground, and mostly contained in separate chapters. The chapter titled 'Physical applications' has a section on perturbation theory, and more generally discusses the importance of symmetries of the Hamiltonian in quantum mechanics. There is also a called chapter 'Applications to atomic and nuclear physics'.

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I can only think of a book: Asim Barut and "can't remember the name" Raczka were the authors. "Group Representations and Applications" is the title. It has been published recently in a revised edition (just do an Amazon search). For anyone desiring rigor in physics, this is the book.

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