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Is a "Group Theory" book different from an "Abstract Algebra" book? –  Matt Gregory Mar 7 '11 at 12:17
I'd take Lang's "Algebra" as an introductory text on any topic in abstract algebra –  shamovic Mar 7 '11 at 13:58
@Matt: Yes, it can be very different. Rotman's "Introduction to the Theory of Groups" is a great introductory (and beyond) Group Theory book, but it would be a pretty lousy introductory Abstract Algebra book... –  Arturo Magidin Mar 7 '11 at 18:41
Good in what sense? If you are asking for a book recommendation, you should describe what criteria you are looking for. –  Willie Wong Sep 8 '11 at 11:24
Better perhaps: go to your local university's mathematics library and dive into the subject. Read and read different books (in my university's mathematics library Group Theory was in the catalog number 23. I'm not sure whether this is international or not) until you find 2-3 that appeal to you more than others (for their simplicity, their organization, their language, notation, etc.), then you can try to read only these ones as a first approach to the subject. –  DonAntonio Feb 23 '13 at 13:31

12 Answers 12

What about the book An introduction to the Theory of Groups by Joseph Rotman?
It is in my opinion a classic.

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I think the group theory part (= first 6 chapters) of Abstract Algebra by Dummit and Foote is quite good. Personally, I dislike Armstrong's book Groups and Symmetry; his style is too informal to my taste, and definitions are hidden in the text.

A concise, clear one is Humprhey's A Course in Group Theory, it gets you quickly to the core of the subject.

For a 'second' course I like the Universitext The Theory of Finite Groups: An Introduction

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I like all your recommendations except your negative opinion on Armstrong's book. I think the definitions are very clear and he has great exercise sets in addition to a very geometric approach to the subject. I certainly wouldn't use it as the sole text in a graduate course,of course. –  Mathemagician1234 Sep 8 '11 at 6:49

If you mean an intro to abstract algebra, A book of abstract algebra by Charles C. Pinter is great. See the reviews on Amazon.

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A very intuitive one is this: N. Carter, Visual Group Theory, MAA 2009.

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Here are some good books:

  • Algebra, Abstract and concrete by Fred Goodman, it's available for download.

  • Abstract Algebra by Dummit and Foote.

  • A first course in Abstract Algebra by John. B Fraleigh.

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In addition to the Rotman and Kurtzwell books suggested below (above?) , I'll make some other suggestions. One of the best introductions I've ever seen to basic group theory is in chapter 1 of I.Herstein's classic Topics In Algebra,2nd edition. This was my introduction to abstract algebra in an honors algebra course and I still think it's one of the truly great all-time algebra books. (I tell graduate students all the time who are worried about their qualifying exams in algebra-take out a copy of Herstein and see how many of his exercises you can do. If you can do most of them without getting stuck,you're ready for your qualifiers.) A great cheap book in Dover paperback for graduate students is John Rose's A Course In Group Theory. This was one of the first books to extensively couch group theory in the language of group actions and it's still one of the best to do that. It covers everything in group theory that doesn't require representation theory. Lastly,a book I had the pleasure of reading and reviewing for the MAA online is I. Martin Issacs' Finite Group Theory. This beautiful,comprehensive text is by a master of the subject and one of the best textbook authors active today. This book differs from the more classical texts in that it's more advanced then most of the others-it begins with the Sylow theorums and assumes basic group theory. As a result,it covers more sophisticated and recent topics then usually found in such texts, such as we meet several results that I doubt have ever appeared in book form before, such as the Chermak-Delgado measure.It's also masterfully written as all Issacs' texts are. It's definitely worth checking out if you're interested in group theory,especially for the very best students.

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Brodkey's theorem appears in Isaacs' earlier textbook: Algebra: A Graduate Course. Also, it is the Chermak-Delgado measure; it is not the "Chernak-Delgado measure". You might also wish to correct this misspelling in your MAA review. Finally, although the following is slightly off-topic, I should also add that "subnormality theory" appears in Derek Robinson's textbook: A Course in the Theory of Groups (contrary to that which is written in your MAA review). –  Amitesh Datta Sep 18 '11 at 1:51
I never read Robinson's text,so I wasn't aware of it's mention in that text. The Brodskey theorum reference was a legitimate error and I've corrected it. And a downvote for a SPELLING ERROR? REALLY?I hope now you can remove the downvotes because Issacs is a really good book and I'd hate people to miss my recommendation because of that. –  Mathemagician1234 Oct 3 '11 at 19:21
Dear Mathemagician1234, thank you very much for correcting this error. I agree that Finite Group Theory by Isaacs is an excellent textbook and that Martin Isaacs is an excellent author; I have thoroughly enjoyed reading his textbooks. However, I downvoted your answer because I felt that your review was not worded carefully enough; please do not take this personally. I recall commenting on another answer of yours on this website and explaining my reasons for downvoting but you deleted your answer and never replied to my comments. Therefore, I do not wish to explain myself again. Best regards –  Amitesh Datta Oct 5 '11 at 3:55
However, I will mention the following criticizm that applies generally to some of your comments and answers; although a few spelling errors or punctuation errors should not result in people downvoting your answers (I certainly would not downvote based on these factors alone), it is important to be very careful in that which you write. In this case, "careful" means that you should confirm beforehand that everything, to the best of your knowledge, is mathematically and factually correct ... –  Amitesh Datta Oct 5 '11 at 4:02
... You are not required to do this but if you do, or at least make some attempt to do so, then I highly doubt that you will receive downvotes. Unfortunately, this is life; I am not the one who dictates these rules but people generally highly regard comments and answers on this website that are mathematically and factually correct; even if they are not, if it is evident that the user in question has made a strong attempt to ensure that this is the case, then people will still appreciate it. No-one is perfect but taking care in what you write on public forums is an important part of life. –  Amitesh Datta Oct 5 '11 at 4:07

I don't know any intro books dedicated solely to t group theory (im sure they exist), but I found Gallian's Abstract Algebra to be incredibly useful as an intro book. The first section is dedicated to groups, and then theres equally good expositions on rings and fields. a little pricey, but really worth it in my opinion. Gallians website also has a ton of great supplementary material

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+1 This was the book we used when I took abstract algebra, and I thought it was very good as an introductory text. –  Tara B Feb 23 '13 at 13:40

My personal opinion is that "Artin, 2nd Ed." is much better than "Dummit & Foote" for an "introductory" text. I think it gives a more intuitive treatment of the material than "D&F."

This can be coupled with Benedict Gross's free video lectures which follow "Artin."


(Although the lectures follow the first edition, the combination provides an outstanding learning experience. A real pleasure.)

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i study group theory from abstract algebra by dummit and foote third ed ,

this text is great , it take you step by step and have many examples , exercises and its explanation is clear ..

it's a great text ..

added : i tried to study from artin but i found it so bad - for me - it's a difficult and the explaination is not clear also it doesn't cover many things in the topic for instance , when it talks about isomorphisms theorems , it show the first theorem only but in dummit the 4 theorems is showed with clear explanation

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I looked at Artin on Amazon where it lets you read sample pages, and I thought it looked terribly old fashioned and wordy, so I agree, that is probably not going to be the book I chose. –  PJW Feb 23 '13 at 13:36
i had artin 1st ed , i studied the first chapter and most of the second chapter , it cover the topics brifely , it give you a little information about any thing ! i don't like this kind of texts .. –  Maths Lover Feb 23 '13 at 13:39

You could try A Book of Abstract Algebra, 2nd ed. by C.C. Pinter.

It is a wonderful Dover book and the first eleven chapters are group theory.

Peruse it on Amazon and see if it fits your needs.

You can also find out which book the class is using and peruse your library.


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You can try any undergraduate abstract algebra book like Topics in Algebra: I. N. Herstein, Algebra by Michael Artin, Abstract Algebra by Gallian,or A First Course in Abstract Algebra by Rotman. Also, there's a book nice book solely dedicated to group theory by Armstrong, Groups and Symmetries.

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Artin is so difficult text in my opinion . –  Maths Lover Feb 23 '13 at 13:32

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