# What are some good group theory references?

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.)

• Not general group theory, but Kaplansky has a lovely little monograph entitled "Infinite Abelian Groups"
– Ryan
Feb 5 at 18:38

Here are a few of my favorite references.

1. For general group theory, my favorite reference is Rotman's book.

2. 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).

3. 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".

4. 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).

5. For Coxeter groups, my favorite references are Bourbaki's volume on the subject and Mike Davis's "The Geometry and Topology of Coxeter Groups".

6. 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).

7. For property (T), I like Bekka, de la Harpe, and Valette's book "Kazhdan's Property (T)".

8. For the symmetric group, I really like G. D. James's "The Representation Theory of the Symmetric Groups".

Robinson's "A Course in the Theory of Groups" is a very good general group theory reference, with a rather extensive bibliography.

A terrific recent addition to the literature I reviewed for the MAA Online is "Finite Group Theory" by I. Martin Isaacs. It contains all the standard material one would expect in a graduate group theory text as well as a number of topics you don't normally see in such texts, like subnormality and the Chernoff measure. It's quite a bit more advanced than the usual group theory texts as well. All of it is presented beautifully with Isaacs' usual authority and scholarship. A GREAT book for anyone interested in group theory with a basic knowledge of algebra.

For older and more standard texts, there's always the old classic by Philip Hall. One of the first of the post-1960's texts and STILL one of the best.

And, of course, there's always John S.Rose's "A Course In Group Theory", available to all in Dover, thankfully. A classic with one of the most complete presentations of the theory of group actions there is.

• I think the book you are referring to was written by Marshall Hall, not Philip Hall (unless you are referring to Philip Hall's Edmonton lectures on nilpotent groups, which are excellent!) May 7, 2010 at 6:05

Serre's Trees has a nice treatment of Bass-Serre theory in the first chapter. This concerns infinite discrete groups. The book displays Serre's usual qualities of very concise writing and an eye for important points.

• If you like to think topologically, I believe it's easier to learn Bass--Serre Theory from Scott and Wall's article 'Topological methods in group theory'. Then again, I actually read Serre first, so maybe I'm mistaken.
– HJRW
May 6, 2010 at 17:28
• It's probably easier to learn Bass-Serre theory from Scott and Wall's article. However, Scott and Wall don't really give any killer applications, while Serre gives an astonishingly slick proof of Ihara's theorem. You should thus probably read both. May 6, 2010 at 18:19

For infinite discrete groups:

• Lyndon & Schupp is authoritative for classical, combinatorial methods.

• Bridson & Haefliger has a lot of material for more geometric classes, like hyperbolic and CAT(0) groups.

As was mentioned Rotman's book is a very good basic book in group theory with lots of exercises.

For finite group theory Isaacs has a relatively new book. I didn't read much from the book, but the little I did, was very nice. Generally, Isaacs is a very good teacher and a writer.

Old fashion references for finite group theory are Huppert's books (the second and third with Blackburn) and Suzuki's books. They are out of print, old fashion and the first of Huppert’s book is in German. But they are encyclopaedic, useful, and popular.

Robinson’s book is a good book especially for infinite group theory, an area which is hard to find in other books.

In my corner of group theory, DDMS, Analytic pro-p groups is standard if you are interested in linear pro-p group, Wilson’s Profinite groups is more general profinite groups theory, and there is also Ribes and Zelesski which I am not familiar with, but I think is more geometric in nature.

A book worth mentioning in my view is Subgroup Growth by Lubotzky and Segal. It contains a lot of group theory and touches on many topics. So by reading it, it is possible to get a good overview of the all area.

• Call me old fashioned, but I also like reading Huppert volume I with its neatly typographied pages, fraktur symbols for groups and subgroups and very careful wording. I regret not to own it, so I have to go to the library to have a look at it. May 9, 2010 at 8:10
• Old fashion is not necessarily bad. It is more a matter of taste. However, I do find fraktur symbols incredibly hard to read. But that might be because Hebrew is my first language. May 10, 2010 at 11:53
• What is DDMS?.. Feb 5 at 20:19
• @LSpice it refers to the last names of the authors of "Analytic pro-$p$-groups": Dixon, du Sautoy, Mann, and Segal. Feb 5 at 20:22

I am surprised Marshall Hall's book: "The Theory of Groups" has not been mentioned. The first 10 chapters of this book cover basic group theory (as much as expected in a graduate course). The last 10 chapters are devoted to advanced group theory. Here, one studies transfers, extenstion theory, representation- and character theory among many other things. It's simply a classic!

• It was mentioned by Andrew L above. Oct 31, 2018 at 11:39
• @JPMcCarthy In fairness, Andrew L didn't really say why it was a good book to use, and he got his Halls mixed up Nov 4, 2018 at 21:00

The book "Linear Algebraic Groups" by Armand Borel and "Linear Algebraic Groups" by James Humphreys are great (and standard) references for the theory of linear algebraic groups. In both of these books, the structure theory of linear algebraic groups uses some algebraic geometry and representation theory.

• These are both great books (Springer's, also by the same title, is too; and Milne's Algebraic groups is arguably even better for a modern student), but probably not what someone asking for general group-theory references has in mind. On the other hand, other answers also get more specialised than I'd expect, so maybe this is what the asker wanted! Feb 5 at 20:18