I am attending a lesson in this semester in Group Theory, in the following special topics. I know that there are similar posts, but in this post I specifically ask to recommend me a combination of well written books or notes, with plenty of worked examples in the following topics:

Group Action on Set and on Group (Permutation Representation, Orbits, Stabilizers, The Orbit-Stabilizer Lemma), Burnside 's Lemma, Transitive Group Action, Group Action by conjugation (normalizer, centralizer), Semidirect product of two groups, dihedral groups, Abelian Groups (Free Abelian Groups with finite rank, Torsion Free Abelian Group, Periodic Abelian Group), The Splitting Theorem in finite generated abelian groups, Sylow Theorems (the method of counting, cycle method), Simple Groups, Small Order Groups, Solvable Groups, Solvability of $S_n$.

PS: I asked for a book combination because I believe that one single book doesn't contain all these topics

Thank you in advance.

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    $\begingroup$ I'd recommend that you not think of the things you've mentioned as "special topics", but as a list of some of the absolutely basic ideas in group theory. Essentially all those things are mentioned in any reasonable abstract algebra book. In particular, do not accidentally think of these things as "obscure" or "esoteric" in any way. They are not at all topics "only for specialists". $\endgroup$ – paul garrett Mar 24 '18 at 22:15
  • $\begingroup$ @paulgarrett Thank you very much Professor for your comment. I wrote "special" because some books that I found have different topics, so I believed that these topics were different from basic. $\endgroup$ – Chris Mar 24 '18 at 22:28

If you are looking for a book which contains a lot of examples I can recommend "A first course in Abstract Algebra" by J. Fraleigh. It has too much text and examples for my taste, but it might be worth to look into. You may look into it here: http://www.vgloop.com/f-/1422977427-302599.pdf it features most of the topics listed.

Another book I have found to suit my preferences better is "Abstract Algebra, Theory and applications" by T. Judson. It presents the same topics in a more precise way than Fraleigh, although It might have less examples. http://abstract.ups.edu/download/aata-20100827.pdf

Last but not least, you should try to get your hands on "Algebra" by S. Lang. Although a bit more complicated than the previous two, but I suggest you should look into them.

  • $\begingroup$ Thank you for your answer. I already posess Fraleigh's book. I agree with you. It's a well written book and I will use it for sure. I ll check and the other book too. $\endgroup$ – Chris Mar 24 '18 at 21:05
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    $\begingroup$ If you are going to take a look at Lang I definitely would recommend using it as a supplement to one of the other books listed here, rather than on its own. It is extremely concise and an excellent reference on the subject (it also has lots of great exercises), but if you are learning abstract algebra for the first time, it would be extremely hard to learn solely from Lang in my opinion. $\endgroup$ – wgrenard Mar 24 '18 at 21:08
  • $\begingroup$ @wgrenard Thank you for your comment and your advice. This is my "second" in group theory but I agree with you. Lang's books are great reference but difficult resource if you are studying topics for first time. $\endgroup$ – Chris Mar 24 '18 at 21:15

Two classic books that contain all the topics listed in your question are:

Robinson, Derek John Scott A course in the theory of groups. Graduate Texts in Mathematics, 80. Springer-Verlag, New York-Berlin, 1982.

Rotman, Joseph J. An introduction to the theory of groups. Fourth edition. Graduate Texts in Mathematics, 148. Springer-Verlag, New York, 1995.

I add the book by Thomas W. Judson Abstract Algebra Theory and Applications available online here.

A good introduction, that I remembered, are the two books by Thomas W. Hungerford for which I studied abstract algebra. The first is very elementary is suitable for anyone who has never seen groups in their life. Abstract Algebra. The second is Algebra. Suitable for anyone who has already had a first group theory course.

  • $\begingroup$ Thank you for your answer. Do you believe that Robinson's or Scott's book is good for first touch in these topics? $\endgroup$ – Chris Mar 24 '18 at 21:20
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    $\begingroup$ @Chris Start with Rotiman's book. $\endgroup$ – Elias Costa Mar 24 '18 at 21:30
  • $\begingroup$ Ok I will start with Rotman. Does "Abstract Algebra" or "Algebra" of Hungerford contain all these? Do you know? $\endgroup$ – Chris Mar 24 '18 at 21:58
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    $\begingroup$ @Chris Everything you asked for in your question can be found in Hungerford's book "Algebra" which is not a book so specialized in group theory but 'algebra' in general. On the other hand the Rotman book is specialized in group theory. Soon the topics you listed in your question are presented in more depth in the Rotman book. All you need to know is this. $\endgroup$ – Elias Costa Mar 25 '18 at 0:09
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    $\begingroup$ @Chris The exercises in Rotman's book are enough. $\endgroup$ – Elias Costa Mar 27 '18 at 20:39

Actually, I know two books, such that each of them contains all those topics (and also many additional information regarding group theory). One of them is "Group Theory" by Alexander Kurosh (however, I do not know whether it is translated to English or not; the language of the original is Russian). The other one is "The Theory of Groups" by Marshall Hall.


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