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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).

Mitchell Faas
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    You can't go wrong with Herstein's Topics in Algebra. – lhf Aug 02 '17 at 12:19
  • 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) – Edward Evans Aug 02 '17 at 12:22
  • Am I correct in noting then, that any (good) abstract algebra book will cover these topics? – Mitchell Faas Aug 02 '17 at 12:47
  • @MitchellFaas Correct, these are all relatively fundamental topics in Group Theory so any book covering Group Theory will contain them. – Edward Evans Aug 02 '17 at 12:50
  • 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. – uniquesolution Aug 02 '17 at 12:59
  • So I'm looking at the books you recommended, as well as Farleigh's book. But I notice that Farleigh's and Dummit and Foote's are both 600+ pages, which I have usually found to only be the case in books that are either overly verbose of lie most of their efforts in developing intuition (whereas I'm a fan of building most of my own intuition). Is this the case here, or is the topic of abstract algebra simply so expansive that so many pages can be filled with dense material? (The old less than an hour per page is probably too fast thing.) – Mitchell Faas Aug 02 '17 at 13:02
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    @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. – Edward Evans Aug 02 '17 at 13:39
  • Personally, I can recommend Armstrong's Groups and Symmetry. ;-) – Hans Lundmark Aug 02 '17 at 14:13

2 Answers2

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I am flabbergasted the following books never got recommended here!

More books are recommended here and here.

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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.

uniquesolution
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  • 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! – Mitchell Faas Aug 02 '17 at 13:04
  • 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. – Edward Evans Aug 02 '17 at 13:19
  • @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. – uniquesolution Aug 02 '17 at 13:22
  • @Meg An overshoot means simply that Dummit and Foote contains lots of other irrelevant information. I prefer to be focused when I study mathematics. – uniquesolution Aug 02 '17 at 13:23
  • @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! :) – Edward Evans Aug 02 '17 at 13:26
  • @uniquesolution From what I can gather it sounds more like a comparison between a regular screwdriver and one with changable bits. Both are very capable of doing the job, but one just has a lot of (irrelevant) extra capabilities. – Mitchell Faas Aug 02 '17 at 13:26
  • @MitchellFaas I wouldn't go as far as to say it contains irrelevant material. It's quite complete (though not even close to something like Serge Lang), but it's also very accessible to the novice reader. – Edward Evans Aug 02 '17 at 13:32
  • @ÍgjøgnumMeg In this case I was aiming at the relevance to the course, which lies at the basis of unique's argument. PS: I did ask a question in the main comments without mentioning your name. I'm not sure if you noticed. – Mitchell Faas Aug 02 '17 at 13:34