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I am currently studying Group Theory from I.N. Herstein's Topics in Algebra.However after studying about 50 pages of it I felt it lacks a bit of geometrical flavour (one of my friends described via email some time back how dihedral groups were treated in his course).He also said that Herstein's treatment is not exactly very modern though it supposedly has some good exercises.

Question: Do I need a supplement to Topics in Algebra that treats the subject in a "modern" way or do I need to change the book(I am skeptical about the second alternative I described)?I would be obliged if someone told me what to do.It seems that I am in a fix.

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You may look at Michael Artin`s book 'Algebra' –  Biswarup Ray Jun 26 '12 at 8:53
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Herstein's book doesn't give much insight into the wide theory of groups, but I think it is a good point to start. Artin's book is more geometrical, but it still contains only the basic theory. For the exercise, you can consider 'Problems in group theory' of Dixon, even if it is not so modern. Do you simply need more examples or do you want to know more? Because in the latter case you probably can simply shift to a true group theory book. –  Vittorio Patriarca Jun 26 '12 at 10:22
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I think every book in Abstract Algebra, especially Groups, has its own smell. If you want to be trained in exercises than doing theorem's proofs, you'd better involve yourself is some books with certain classified problems. Some of them like what Vittorio noted are based on this approach. Dixon is good but rather old. But it gives you much insight. –  Babak S. Jun 26 '12 at 13:59
    
Some like "Algebra through Practice: Groups" by T. S. Blyth, E. F. Robertson does the same. There is another book, I remember, A First course in Abstract Algebra by Fraleigh. It is the old one as well, but throughout the book; he managed good multiple choices which make your knowledge about Abstract Algebra solid. :-) –  Babak S. Jun 26 '12 at 14:00
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I would say tat Herstin is a great algebra book, but it doesn't go very far into Group Theory. A good (but fast) text for Group Theory is that of M. Aschbacher –  Geoff Robinson Jul 14 '12 at 23:06

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All of the books named in the comments are good choices. Probably look at as many as you can, and decide which one(s) you would like to work through.

One book not listed is Dummit & Foote's Abstract Algebra. This book is a standard intro algebra book, for both undergraduate and graduate courses. It is fairly encyclopedic, so you would have a lot to work with. I also like how in its treatment of groups emphasizes group actions, a very fruitful idea that is left out of some books, for example Gallian's Contemporary Abstract Algebra.

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I first learned algebra from Herstein in Honors Algebra as an undergraduate, so the book will always have a special place in my heart. Is it old fashioned? I dunno if at this level it really makes that much of a difference. No, it doesn't have any category theory or homological algebra and no, it doesn't have a geometric flair. What it does have is the sophisticated viewpoint and enormous clarity of one of the last century's true masters of the subject and perhaps the single best set of exercises ever assembled for an undergraduate mathematics textbook. When a graduate student is scared to take his qualifier in algebra, I give him some very simple advice: I tell him to get a copy of the second edition of Herstein and to try to do all the exercises. If he can do 95 percent of them, he's ready for the exam. Period.

So to be honest, I don't think it's really necessary when you're first learning algebra to get a more "modern" take on it. It's more important to develop a strong foundation in the basics and Herstein will certainly do that. If you want to look at a book that focuses on the more geometrical aspects as a complement to Herstein at about the same level, the classic by Micheal Artin is a very good choice. (Make sure you get the second edition,the first is very poorly organized in comparison!) Another terrific book you can look at is E.B.Vinberg's A Course In Algebra, which is my single favorite reference on basic algebra. You'll find it as concrete as Artin with literally thousands of examples, and it's beautifully written and much gentler then either of the others while still building to a very high level. I think it may be just what you're looking for as a complement to Herstein.

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Can you elaborate on how the second edition of Artin's Algebra is better organized than the first edition? –  littleO Nov 27 '12 at 10:48

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