I'm looking for a good (intoductionary) book on group theory that treats (at least) the following material:

The axioms of groups, commutativity, symmetrical groups, permutation groups; Cayley's theorem, matrix groups, order of a group, homomorphisms between groups, isomorphisms between groups, undergroups; Lagrange's theorem, quotient groups; normal undergroups, homomorphesm, kernal, image, isomorphism theorems, group actions; the orbit stabilisator theorem, direct products of groups, the cauchy theorem, sylowgroups; Sylow's theorem and "free" group.

The book the course recommends is: Armstrong, M.'s Groups and symmetry.

That said, I've found that I usually get much better book recommendations by asking around on here. When it comes to personal preference, I enjoyed Kunze's Linear algebra and Rudin's Mathematical Analysis. I prefer books not be too verbose in their approach to the subject and am okay with losing (some of) a proof's detail in favour of a clear portrayal of the important steps (I believe Rudin does this really well).

  • $\begingroup$ You can't go wrong with Herstein's Topics in Algebra. $\endgroup$ – lhf Aug 2 '17 at 12:19
  • $\begingroup$ Dummit and Foote's Abstract Algebra is what I used/use (since it covers topics from basic theory all the way to graduate level topics in algebra) $\endgroup$ – Edward Evans Aug 2 '17 at 12:22
  • $\begingroup$ Am I correct in noting then, that any (good) abstract algebra book will cover these topics? $\endgroup$ – Mitchell Faas Aug 2 '17 at 12:47
  • $\begingroup$ @MitchellFaas Correct, these are all relatively fundamental topics in Group Theory so any book covering Group Theory will contain them. $\endgroup$ – Edward Evans Aug 2 '17 at 12:50
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    $\begingroup$ @MitchellFaas Abstract Algebra encompasses more than just group theory, the text I'm recommending has about 230 pages on group theory, and then more on rings/fields and then some on more specialised topics in algebra. It's not as dense as Serge Lang (which is very, very abstract and very, very dense, I use it for reference only), but it's detailed enough to be complete. It's nicely written too so it doesn't get boring. Most definitions/theorems are accompanied by 5-10 examples to illustrate the concepts. $\endgroup$ – Edward Evans Aug 2 '17 at 13:39

You don't need a book in abstract algebra. You need an introductory book that deals exclusibely with group theory. Dummit and Foote is an overshoot. Rotman's "An introduction to the theory of groups" or Robinson's "A course in the theory of groups" are good books for your needs.

  • $\begingroup$ What do you mean that it's an overshoot? I'm personally not averse to books that cover (much) more material than needed, as I find it more important to look for the best books available in the subject. If they extend further, I may only benefit from that in future courses or as reference. So some elaboration on your statements would be hugely appreciated! $\endgroup$ – Mitchell Faas Aug 2 '17 at 13:04
  • $\begingroup$ Just because the book covers more than just Group Theory, doesn't make it an overshoot... Dummit and Foote has a LOT of exercises and gives a LOT of examples for concepts, which I found incredibly useful. $\endgroup$ – Edward Evans Aug 2 '17 at 13:19
  • $\begingroup$ @MitchellFaas When you need to drive an nail in the wall, do you bring a 10 pound hammer? No, you don't, and when someone tells you that a 10 pound hammer is an overshoot for driving a nail into the wall, do you nudge them to elaborate? I hope not. You want to use D&F? By all means do. You asked a question, I gave my answer. $\endgroup$ – uniquesolution Aug 2 '17 at 13:22
  • $\begingroup$ @Meg An overshoot means simply that Dummit and Foote contains lots of other irrelevant information. I prefer to be focused when I study mathematics. $\endgroup$ – uniquesolution Aug 2 '17 at 13:23
  • $\begingroup$ @uniquesolution Okay, that's your opinion and is to be respected, Dummit and Foote is a good text with good exposition though so I stand by my recommendation! :) $\endgroup$ – Edward Evans Aug 2 '17 at 13:26

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