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Which textbook is good for introductory group theory?

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    $\begingroup$ Is a "Group Theory" book different from an "Abstract Algebra" book? $\endgroup$ Mar 7, 2011 at 12:17
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    $\begingroup$ I'd take Lang's "Algebra" as an introductory text on any topic in abstract algebra $\endgroup$
    – shamovic
    Mar 7, 2011 at 13:58
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    $\begingroup$ @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... $\endgroup$ Mar 7, 2011 at 18:41
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    $\begingroup$ Good in what sense? If you are asking for a book recommendation, you should describe what criteria you are looking for. $\endgroup$ Sep 8, 2011 at 11:24
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    $\begingroup$ 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. $\endgroup$
    – DonAntonio
    Feb 23, 2013 at 13:31

14 Answers 14

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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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    $\begingroup$ Title aside, do you think it is really suitable as an introduction for an undergradute? $\endgroup$ Apr 9, 2015 at 22:19
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    $\begingroup$ Furthermore, according to several of the Amazon reviews, e.g. one by Adam Glesser, the 1995 edition/printing (still the latest?) has many errors ranging from typos to undecipherable proofs. This has also been discussed on M.SE math.stackexchange.com/questions/302508/… I can't find an errata for the book [written by its author]. Rotman has some errata pages for his other [more recent] books on math.uiuc.edu/~rotman $\endgroup$ Apr 11, 2015 at 23:02
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    $\begingroup$ There's 2nd party errata for Rotman's Intro to Groups at people.ds.cam.ac.uk/mg262/rotman%20group%20theory,%20copy.pdf, found in the other M.SE thread. This errata is 45 pages long! Granted this is due to its discussion-oriented form. So at least for the introductory material, if you adore Rotman's style, you're probably better off reading it from his Advanced Modern Algebra instead. $\endgroup$ Apr 11, 2015 at 23:08
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    $\begingroup$ One may be less than convinced by many considerations in the text referred to as "errata" in a comment above. The text is found on the website of Mohan Ganesalingam, author of a PhD titled "A Language for Mathematics", and this specific viewpoint is apparent in the comments to Rotman's book. $\endgroup$
    – Did
    Apr 12, 2015 at 7:18
  • $\begingroup$ I don't see why an introduction cannot contain errors, nor how that relates to its being suitable for an undergraduate. And it is in my humble opinion a classic. Also, thanks for the recommendation of Advanced Modern Algebra. $\endgroup$
    – awllower
    Apr 12, 2015 at 12:55
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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 by Kurzweil and Stellmacher.

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  • $\begingroup$ 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. $\endgroup$ Sep 8, 2011 at 6:49
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    $\begingroup$ @Mathemagician1234: My quick test for deciding if a book on geometry and groups is [not] precise enough is whether you can deduce from its definitions why there are more fundamental Frieze patterns than there are abstract groups underlying them. Armstrong's book kinda fails this with "Two frieze groups should be thought of as equivalent if they are isomorphic via an isomorphism which sends translations to translations, rotations to rotations, reflections to reflections, and glides to glides." That's basically the solution to an unstated problem; it is taken as definition in Armstrong's book. $\endgroup$ Apr 9, 2015 at 22:37
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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 than most of the others-it begins with the Sylow theorems and assumes basic group theory. As a result, it covers more sophisticated and recent topics than 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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    $\begingroup$ 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). $\endgroup$ Sep 18, 2011 at 1:51
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    $\begingroup$ 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. $\endgroup$ Oct 3, 2011 at 19:21
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    $\begingroup$ 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 $\endgroup$ Oct 5, 2011 at 3:55
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    $\begingroup$ 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 ... $\endgroup$ Oct 5, 2011 at 4:02
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    $\begingroup$ ... 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. $\endgroup$ Oct 5, 2011 at 4:07
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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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  • $\begingroup$ IMHO, Pinter's book can be described by two "coordinates". One is that it basically introduces just enough group theory so it can get to Galois theory; a terseness [of topics] which I often find irritating. It doesn't discuss group actions for example, nor does it mention semidirect product (normal subgroups are there, of course). [continutes in the next comment] $\endgroup$ Apr 9, 2015 at 22:17
  • $\begingroup$ The other "coordinate", somewhat correlated with the first one, is that the book will often spend time gently introducing basic (set theory) notions like a partition etc. Those who need these concepts introduced (apparently a lot of the Amazon customers) will appreciate this book; others, not so much. $\endgroup$ Apr 9, 2015 at 22:18
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    $\begingroup$ I've had bit more time to waste by looking at Pinter's book, and I have extra comments: I was wrong about actions not being in the book at all, they are mentioned in an exercise. Which actually brings me to the main worry/criticism of his book: "advanced" material like the isomorphism theorems (except the 1st one) are left as exercise to the reader, i.e. stated but not proved. I seriously doubt that struggling students (needing a chapter/section on what functions and/or partitions are) are going to be able to prove such results as the isomorphism theorems themselves... $\endgroup$ Apr 11, 2015 at 1:27
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    $\begingroup$ [slightly over the word limit in the previous comment]: And the same goes for the Cauchy and Sylow theorems... they're also left as exercises. $\endgroup$ Apr 11, 2015 at 1:31
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A very intuitive one is this: N. Carter, Visual Group Theory, MAA 2009.

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  • $\begingroup$ The weak point of Carter's book is that its text if often enough verbose but muddy. Consider the long question and long answer it generated here as well as the long back-and-forth for a rather simple issue (Cayley diagrams of semigroups). Also, the latter part of the book has more significant errors, especially in the Galois theory chapter. A large part of the web errata for the book addresses that. $\endgroup$ Apr 5, 2015 at 13:06
  • $\begingroup$ This class note has summarized it up to the homomorphism chapter. But it skips Fig. 7.31 on the normalizer of the original book, which I find very intuitive. If you have some knowledge on group beforehand, you can skim the class note for images only $\endgroup$
    – Ooker
    Oct 23, 2017 at 23:53
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I don't know any intro books dedicated solely to t group theory (im sure they exist), but I found Gallian's Contemporary 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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    $\begingroup$ +1 This was the book we used when I took abstract algebra, and I thought it was very good as an introductory text. $\endgroup$
    – Tara B
    Feb 23, 2013 at 13:40
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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."

http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra

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

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  • $\begingroup$ I honestly found Artin's book rather hard (in the sense of annoying) to follow because of the huge number of back-references in formulaic manner, along the lines of proof of theorem 6.x.yz making backref to formula 5.a.b.cd. Good math books manage to minimize this kind of tedium. An ebook with click-able back-references would be a significant improvement in this regard, but such an edition doesn't seem available. Artin's book also calls group action (on a set) just "operation" which I found pretty annoying and occasionally confusing. $\endgroup$ Apr 1, 2015 at 18:55
  • $\begingroup$ @RespawnedFluff Thanks for your opinion. Yes, it is a pain to have to flip around. Unfortunately I find a lot of kindle math books have lots of typos due to notational issues. Sometimes I have spent too much time trying to untangle them, and a look at a print edition comes to the rescue. However, I think we would both agree that the prices of math book are obscene, and "Artin" in particular. Regards, $\endgroup$
    – user12802
    Apr 1, 2015 at 20:15
  • $\begingroup$ I'm voting this up just for mentioning Gross's video lectures. They are a treat! $\endgroup$
    – user940
    Nov 30, 2016 at 16:31
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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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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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  • $\begingroup$ Artin is so difficult text in my opinion . $\endgroup$
    – FNH
    Feb 23, 2013 at 13:32
  • $\begingroup$ I feel I should mention that compared to his group theory book or to his graduate algebra book (titled Advanced Modern Algebra), Rotman's undergraduate text A First Course in Abstract Algebra had rather more negative reviews, especially on Amazon (rather than in more formal venues like journals.) It seems the main complaint was that proofs were not sufficiently detailed for an undergraduate textbook. $\endgroup$ Apr 11, 2015 at 19:28
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Antonio Machì, Groups: An Introduction to Ideas and Methods of the Theory of Groups.

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    $\begingroup$ This is another a relatively new book; 2012 it seems. It doesn't seem badly written from a couple of pages I've read in a preview. But it would be nice if you have more impressions about it to share, if you've read more of it. $\endgroup$ Apr 9, 2015 at 22:00
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Since it was not cited so far, I recommend J.S. Milne's Group Theory.

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  • $\begingroup$ This a bit condensed, but in the category of free texts it has little competition. Actually it has some competition in Thomas W. Judson's Abstract Algebra: Theory and Applications; Judson's book is more introductory. $\endgroup$ Apr 11, 2015 at 18:50
  • $\begingroup$ Judson spends about half of his books length (~200 of 400 pages) covering groups. Another free book is Goodman's, discussed in another post here. From a quick look, the coverage of groups in Judson vs. Goodman is very similar both in terms of length and depth of exposition. Goodman eventually gets to more advanced topics like modules, which aren't in Judson. $\endgroup$ Apr 11, 2015 at 19:16
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I strongly recommend Frederick M. Goodman, Algebra: Abstract and Concrete

Also it is a free source and in my opinion the best introduction to groups and fields that a student could possibly have. Prerequisites are not much except some experience and patience to work through problems (there are many good problems in this text) and read and understand the many proofs. A theme in this book is to use simple symmetries and rotations to introduce the axiomatic concept of a group which in my opinion can be a little difficult to grasp with just it's definition alone. Group homomorphisms and isomorphism, measure of groups commutative through commutator group, cosets, quotient groups and isomorphism theorems, Lagrange' theorem about order of groups.The theories they introduce are then easy to grasp. Finally an introduction to product groups, then on to generalize many group theoretic structures to that of fields. Many examples are from linear algebra. I think this book would get a student well on there way to a healthy knowledge of group theory. After completion of this book I think you would feel ready to go on to study abstract algebras in general and on to the study of universal algebra. Unfortunately I can not provide a reference for these fields that can compare to the wonderful methods used in this book.

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  • $\begingroup$ It looks pretty decent. A comparable [free] resource as far as intro to group theory goes is Thomas W. Judson's Abstract Algebra: Theory and Applications. From a quick look, the coverage of groups in Judson vs. Goodman is very similar both in terms of length and depth of exposition. Goodman eventually gets to more advanced topics like modules, which aren't in Judson's book. For more advanced topics in group theory per se, there are J.S. Milne's notes (mentioned in another post here), which have a more condensed/graduate approach. $\endgroup$ Apr 11, 2015 at 19:19
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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.

Enjoy!

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    $\begingroup$ To add something more concrete here, Pinter's book can be described by two "coordinates". One is that it basically introduces just enough group theory so it can get to Galois theory. Which I often find irritating. It doesn't discuss group actions for example, nor does it mention semidirect product (normal subgroups are there, of course). The other "coordinate", somewhat correlated with the first one, is that the book will often spend time gently introducing basic (set theory) notions like a partition etc. Those who need these concepts introduced will appreciate this book; others not so much. $\endgroup$ Apr 9, 2015 at 22:12
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    $\begingroup$ This books was talked about up there. Why was this posted again, seriously. These all should be skimmed at least before one were to post. $\endgroup$
    – smokeypeat
    Jan 31, 2016 at 20:16
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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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    $\begingroup$ 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. $\endgroup$
    – PJW
    Feb 23, 2013 at 13:36
  • $\begingroup$ 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 .. $\endgroup$
    – FNH
    Feb 23, 2013 at 13:39

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