I am beginning to learn group theory (specifically finite groups) and I’m wondering which textbooks can help me.
So any suggestions for introductory texts?
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There are quite a few (very) good textbooks on finite groups out there. My favourite is Isaacs' "Finite Group Theory", but I wouldn't recommend it as a first textbook. Robinson's "A Course in the Theory of Groups" and Rose's "A Course on Group Theory" are both excellent.
In my opinion, though, the best book to read as a first when it comes to group theory is Smith's and Tabachnikova's "Topics in Group Theory".
My introduction to groups (and algebra in general) was I.N. Herstein's Topics in Algebra. It begins with the theory of groups (covering what I would regard as the essential basics), but also covers rings, fields, vector spaces, and linear transformations. It's short, well written, and has a lot of good exercises.
I also really like Michael Artin's Algebra, which again is an introductory algebra textbook, but it includes quite a lot of good material on groups. I would say it's a bit more comprehensive than Herstein, and also very well written of course.
Lastly, I've gotten a lot of mileage out of Milne's course notes, which between them cover almost all of the algebra I've ever needed to know. It's been a while since I've read his group theory notes, but they're free on his website, so it's worth checking out.