3

My background is in computer science, specifically software engineering, and not really math heavy. I know the basics of calculus (the Thomas book) linear Algebra (Strang), and some Discrete Math, Graph Theory, Complexity and Algorithms and that's about it.

I was recently working on proving that the 8-slide puzzle states are divided in to 2 disjoint sets (AI exercise). I had no clue that permutation groups were such a large field of study on their own (I can't even understand the terminology in wikipedia articles, orbits? Cayley tables?), or that their study was Abstract Algebra stuff.

My question is, what is the learning path, book-wise or just topic wise (and I'll find the books later) for someone like me, to understand the basics of Group Theory (taking into account that I don't even know what it really is) and permutation groups more specifically?

P.S. I'm not asking because I want to solve my exercise (this was simple) it just seems a fascinating topic.

amWhy
  • 210,739
foofootos
  • 33

3 Answers3

5

Another book which I find very accessible is Fraleigh's A First Course in Abstract Algebra, which would be a great start with which to begin, and requires no more in the way of prerequisites than you've already covered. You wouldn't need to cover all the material in Fraleigh's text, if you're interested primarily in group theory: the basics you'll need for getting an overview of groups (including permutation groups) is covered in the first half of the text, with some supplemental sections on groups included (among other topics) in the latter half. I've had a lot of success with using it in an undergraduate's first course in abstract algebra/modern algebra.

It certainly wouldn't hurt to work through Fraleigh in conjunction with Dummit and Foote. They do a great job with group theory, and it is an excellent text, overall.

See this post: For more suggestions on abstract algebra texts, at varying levels of difficulty.

At any rate, once you have a basic "lay of the land" and familiarity with abstract algebra, with a focus on groups, then check into An introduction to the Theory of Groups by Joseph Rotman. It's a classic.

Community
  • 1
amWhy
  • 210,739
  • One text that is not included in that link which I like: M. Armstrong's Groups and Symmetry. It approaches group theory mostly via motivating examples in Euclidean geometry, and thus I found it a pleasure to read. It only deals with group theory, and there are more complete treatments, but it'll get you quickly acquainted with many important groups including permutation groups. – Gyu Eun Lee Jul 12 '13 at 03:15
  • Thanks guys, going to the library to get Fraleigh's book for now – foofootos Jul 12 '13 at 13:30
  • Very nice suggestion! Personally I found Dummit and Foote's book excellent. –  Apr 20 '14 at 12:17
4

I saw that @amWhy posted a link including lots of references, but If you want to have a concrete base in Permutation Groups, I strongly suggest you to see the following books as well. :-)

  • Finite Permutation Groups by Helmut Wielandt.

  • Permutation Groups by John D. Dixon, Brian Mortimer.

Mikasa
  • 67,942
  • There is a text for permutation groups written by the man who you have dedicated your answers and hints for - Peter Cameron - . if you know this text , Do you recommend it ? – FNH Jul 12 '13 at 13:09
  • 1
    @MathsLover: Yes yes it is a great one however I read Wielandt and some chapters of the second book. I prefer Wielandt. – Mikasa Jul 12 '13 at 18:24
2

Hungerford's Abstract Algebra: An Introduction has a nice treatment of the theory and talks about permutation groups. It is written at a pretty easy level. When I first read it all I had under my belt was calculus and linear algebra. I found it very accessible. Artin's Algebra is a great book which I have found very useful in many areas including computer science. You may find it a little advanced though. You can probably just jump into basic group theory without reading anything else first. If you want to develop a background that may be beneficial you can read some number theory and combinatorics but I don't think this is necessary.

Wintermute
  • 3,930