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I've read the basics of this branch and I found it extremely interesing, and I would really love to learn more about it.

I want to study as much as I can on my own, as my course doesn't have group theory, unfortunately.

What would be a good (ground-up level) introductory book, and then a mid-level group theory book?

Thanks so much in advance.

E: I'm taking a number theory course at the moment. I haven't taken any other math courses yet, although I've some studied calculus, real analysis and linear algebra on my own.

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  • $\begingroup$ You should add more context to get better recommendations. Have you studied any number theory before? What are some recent math courses (with books if possible) you've taken? $\endgroup$
    – aes
    Apr 28, 2015 at 3:58
  • $\begingroup$ @aes I'm studying number theory right now. I haven't taken any other math courses yet, although I've studied calculus, real analysis and linear algebra on my own. I'll add that in my question. Thanks! $\endgroup$ Apr 28, 2015 at 4:01

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In my opinion, Artin's is a pretty decent introductory text; I learned the very basics of abstract algebra predominantly from that book. The first chapter focuses on matrices, which was actually very good for me because my linear algebra course had focused on the abstract picture of linear algebra, and spent very little time with matrices. Artin does have a few linguistic oddities; off the top of my head, the use of the phrase "law of composition" for a binary operation, "Group operations" instead of group actions. Overall, though, I'd say it's a good comprehensive text to get you started.

For a text that has more meat to it, but is still well grounded for the beginner, I'd go for Dummit and Foote. If you're feeling ambitious.

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I would go with Beachy and Blair. Just read some definitions and know statements of theorem. Work as many problems as you can, pass an example through each proof to see how it works and go from there. I would not start with really high level literature, this is often very discouraging. If you want more of a challenge just occasionally check the group theory tag to see if you can provide insight or even better provide solutions, hints, etc.

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I personally learned algebra from Herstein's Topics in Algebra, and found it a very readable book with many good exercises. The notation is definitely a little nonstandard in places, and it is certainly an introductory course, but a very good book.

Aside from that, I've also read a fair bit of Artin's textbook, and I've found it also to be very good, I haven't done too many of the exercises myself, but they seem to be pretty good as well from the ones I have done.

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If you're learning on your own, you might want a book with answers, such as Basic Abstract Algebra by Ash or Topics in Group Theory by Smith & Tabachnikova. If you find these go too fast at first, you could start with A Book of Abstract Algebra by Pinter or A First Course in Abstract Algebra by Fraleigh, or something along those lines.

You can supplement the above with Carter's Visual Group Theory and/or Alcock's How to Think about Abstract Algebra. The former can help with "seeing" what's going on, and the latter is a gentle introduction containing advice on how to study abstract algebra, including an explanation of some of the basics of group theory and a small taste of rings.

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    $\begingroup$ I'd like to add the beautiful lecture series by Prof. Macauley which is freely available on youtube, since you've mentioned Carter's book :). He uses Carter's Visual Group Theory as a basis. Very nice for beginners! $\endgroup$ Feb 4, 2022 at 12:57
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    $\begingroup$ Basic Abstract Algebra by Robert B. Ash couldn't be recommended often enough. $\endgroup$
    – Zest
    Feb 4, 2022 at 13:14

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