I'm curious about what books people use for a group theory reference. I don't currently own a dedicated group theory book, and I think I'd find such a book very helpful in my research. What is your favorite book on group theory? Please tell us why you like it — and what sort of groups it focuses on (finite? discrete? finitely generated? etc.)
(For my part, I'm interested mainly in discrete, finitely generated groups, but I enjoy the "flavor" of general group theory books more than combinatorial group theory books.)
Best Answer
Here are a few of my favorite references.
For general group theory, my favorite reference is Rotman's book.
For finite groups, my favorite reference book is Carter's "Simple Groups of Lie Type", which probably reflects the fact that most of the finite groups I have to deal with are things like $\text{SL}_n(\mathbb{Z}/p\mathbb{Z})$. However, when I need info on the representation theory of these groups, I end up turning to Steinberg's lecture notes (alas, not in print).
For infinite groups like $\text{SL}_n(R)$ with $R$ a ring, my favorite reference is Hahn and O'Meara's "The Classical Groups and K-Theory". Another important reference here is Bass's book "Algebraic K-Theory".
For arithmetic groups (here there is some overlap with answer 3), I like Dave Witte Morris's book on the subject (it's not in print yet, but it is available on his webpage).
For Coxeter groups, my favorite references are Bourbaki's volume on the subject and Mike Davis's "The Geometry and Topology of Coxeter Groups".
For geometric group theory, in addition to the wonderful book of Bridson and Haefliger that Henry mentioned, I like Pierre de la Harpe's book on the subject (mostly for the amazing bibliography).
For property (T), I like Bekka, de la Harpe, and Valette's book "Kazhdan's Property (T)".
For the symmetric group, I really like G. D. James's "The Representation Theory of the Symmetric Groups".