Choosing an advanced group theory text: concerns

Question

In this question, An Introduction to the Theory of Groups by Rotman is recommended twice as a good second-course group theory text. However, after reading the reviews here, and seeing this pdf of what seems to be corrections to Rotman, I am pretty concerned about the apparently many errors in the text. While some errors and their corrections may be pretty self-evident, I would hate to constantly be worrying about developing some misconception as a result of some fact that isn't presented quite as it should be.

To those who have read An Introduction to the Theory of Groups by Rotman: do you feel the errors are a serious concern/disruption to the flow of the text, or is the book still pretty navigable?

Also recommended in the question linked to above was A Course in the Theory of Groups by Robinson. I really like the look of this text, based on the table of contents and the group-theoretic concepts that really interest me.

The Robinson text is said (in the linked-to question) to 'move pretty quickly into deeper waters.' My background consists of basic group theory, ring theory, field theory, a little Galois theory, linear algebra, topology, real analysis, and graph theory (I have seen a decent amount of material and have been self-studying for a few years now).

Do you think that I can reasonably wade into these deeper waters without going in over my head and drowning? I realize this is a little subjective since no one knows my exact abilities, but I love group theory and am definitely willing to put some real effort into it.

Thanks in advance for the advice.

Answer 1

Personally I think Robinson would be a terrible book to learn with. It is a great book, but it's a great reference book, or at most a good book for getting into research level group theory (mostly in the second half). It is incredibly dense and offers very little in the way of helpful exposition. As for the exercises, the difficulty runs from very easy to impossibly hard (which is not a bad thing).

Rotman is a good book. I haven't noticed any errors that weren't obvious after first glance. The exercises are a good level of difficulty for people just moving into advanced group theory - challenging enough to be fun, yet not so hard that you become exhausted and lose your motivation to continue. The exposition is pretty good too. I like it. If you're into finite group theory, check out Isaacs' book by that name - it is the best I've read across the board.

EDIT: Also, I took a look at the error log you posted for Rotman, and the majority of the changes are very nitpicky, such as replacing "one says that" with "we say that" in definitions.

Answer 2

Books I love, approximately in the order I could understand them (ignoring publication dates; also you might notice groups are finite unless otherwise specified):

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• Rotman's Group Theory (actually I don't really like it anymore, but I loved it back in the day)
• Hall's Theory of Groups (awesome, low key, some deep results)

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• Isaacs's Finite Group Theory
• Alperin–Bell Groups and Representations (for GL and Sylow; short)
• Wehrfritz's Second Course (for solvable ideas)
• Suzuki's Group Theory (elementary, but covers some serious matrial)

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• Robinson's Theory of Groups (working up to infinite soluble groups and finiteness conditions)
• Hans–Kurzweil Theory of Finite Groups (clean, crisp; was the best until Isaacs's; still has best description of the transfer homomorphism, but Isaacs gives a better description of transfer itelf)
• Gorenstein's Finite Groups (classic; you can start it earlier, but the maturity level required is uneven; probably need Isaacs's CTFG first)
• Aschbacher's Finite Group Theory (clean, crisp; if you understand a chapter, then its amazing, but some chapters you might not get, and there is not much there to help you)

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• Doerk–Hawkes Finite Soluble Groups (the first two background chapters are actually the first three volumes of Endliche Gruppe condensed)
• Huppert's Endliche Gruppe
• P. Hall's Collected Works (well, I only read the finite ones)

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Also try to read Isaacs's Character Theory of Finite Groups as soon as you can. I suspect it should be fine after Rotman or Hall. After that, James–Liebeck's Representations and Characters of Groups continues the same path. You can get pretty far without character theory, but it's just silly not to bask in its awesomeness.

Wilson's Finite Simple Groups, Carter's Simple Groups of Lie Type, Holt–Plesken Perfect groups, Leedham-Greene–McKay Structure of Groups of Prime Power Order, and Malle–Testerman Linear Algebraic Groups and Finite Groups of Lie Type are amazing too, but you probably know if you need them.

At any rate, this should give you an idea of the wonderful things you can read. :-)

Answer 3

I have the perfect recommendation for a student at your level, Alex. I reviewed the amazing graduate text, Finite Group Theory by I. Martin Isaacs, for the MAA online reviews several years ago and I still think it's the best currently existing second exposure to group theory. It's an extremely challenging book and students shouldn't even think about reading it without a strong undergraduate background in algebra, such as year long honors course based on Artin or Herstein. You certainly seem to have more then enough background to handle,so I'm happily recommending it. It's beautifully written by one of the top researchers in the field and contains many topics you won't find in standard texts.

To quote from my original review:

Each chapter comes with a freight car full of substantial exercises, ranging in difficulty from trivial to research level, many of them defining aspects of group theory not covered in the text proper, such as the Frattini subgroup, elementary abelian groups, the quasiquaternion and generalized quaternion groups, extraspecial groups, supersolvable groups and much, much more — some of which are later used in the text proper. The book also has the one telling characteristic of a text written by an active researcher in the field — the material covered reaches much closer to the research frontier then is usual. This is particularly clear in the chapters on subnormality and transfer theory, which contain many fairly recent results.

The book is amazingly clean. I couldn’t find a single error. But the very best thing Isaacs brings to this book is the same thing he brings to all his textbooks — his wonderful style. Definitions, theorems and associated results are presented in a remarkably well organized, coherent manner, all in the author’s terrific lively prose. In this regard, the book reads at times less like a textbook and more like a novel on the great narrative of the story of the development of finite group theory over the last twelve decades. The running theme unifying all these results in the narrative is the great accomplishment of the classification of finite simple groups. The flowing, eclectic style certainly conveys the vast love of the author for his chosen specialty and his great desire to set others on the same path.

Get this book. You'll thank me later, I promise.