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Which textbook is good for introductory group theory?

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    The problem with this question is that "introductory group theory" can mean two or three (or even four) things: (1) the material on groups that generally gets intro'd to undergraduates in a first course on algebra. Usually ends with Sylow theorems, but no characters etc. (2, 3) The material that one can find in a 2nd-ish undergraduate course covering groups alone, usually finite groups and or in the 1st graduate year material on groups. (4) The material that physicists or some computer scientists need (e.g. in computer vision); this is 90% Lie groups. I assume it's not (4), but otherwise...2015-04-09

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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    @Did: Thanks that explains a lot about that rather strange document.2015-04-12
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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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    @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.2015-04-09
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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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    ... 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.2011-10-05
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A very intuitive one is this: N. Carter, Visual Group Theory, MAA 2009.

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    This [class note](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwi4iJTH94fXAhUGxLwKHdpCCpcQFggnMAA&url=http%3A%2F%2Fsums.ucsd.edu%2Fstatic%2Fresources%2Fclass%2F100A-B-C%2FVisual%2520Group%2520Theory.pdf&usg=AOvVaw31kdnFYDgw9Mb6NN5pGUVZ) 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 only2017-10-23
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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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    [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.2015-04-11
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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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    I'm voting this up just for mentioning Gross's video lectures. They are a treat!2016-11-30
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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 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.2013-02-23
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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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Antonio Machì, Groups: An Introduction to Ideas and Methods of the Theory of Groups.

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    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.2015-04-09
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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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    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.2015-04-11
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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 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 ..2013-02-23
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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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    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.2016-01-31
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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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    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](http://abstract.ups.edu/). 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.2015-04-11
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Since it was not cited so far, I recommend J.S. Milne's Group Theory.

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    Judson spends about half of his books len$g$th (~200 o$f$ 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.2015-04-11