[SOLVED] Group Theory For Dummies

climbhi

I've become interested in learning about Group Theory. I don't know too much but I see it spring up all over the place and would just like to know what it is about and some of the basics. Could some one please point me in the direction of a good resource that wouldn't be too far over my head? Thanks.

BoulderHead

Oops, I thought you said 'group therapy for dummies'. I guess I can't help after all. [:D]

climbhi

Well perhaps I need some of that too...

rutwig

Good (introductory) references are:

M. A. Armstrong. Groups and symmetry. Springer Verlag 1988.

J. Rotman. An introduction to the theory of groups, Sprnger Verlag 1995.

J. D. Dixon, Problems in Group Theory. New York: Dover, 1973.

R. Mirman. Group Theory: An Intuitive Approach. World Scientific, 1995.

damgo

Hmm, I used Dummit&Foote "Abstract Algebra" as an introductory textbook, and found it to be excellent. Though the emphasis is on mathematics rather than the physics applications (Lie groups, representations, etc). The nice thing about pure group theory is it requires basically zero prerequisites.

chroot

I'll also recommend the Schaum's Outline of Group Theory. It doesn't specifically cover some of the more physically interesting topics such as groups of 3x3 matrices, but it gives you all of the tools necessary to understand just about any group-theoretical system.

- Warren

Tom Mattson

Yep, and the one called Abstract Algebra, too. All in all, a 30 dollar committment.

climbhi

Thanks for the replies, does anyone know of a good online (read free) source? I'm kind of interested in group theory to see how it relates to QM and what not, but also just for pure math. Would the Schaubs outlines work well for both if there is no good free source available?

rutwig

Here a book that has been a quite interest source for many physicists/


http://www.cns.gatech.edu/GroupTheory/index.html

Tom Mattson

There are many free sources available, but I recommend the Schaum's outlines anyway, because they are loaded with solved examples and exercises with answers.

marcus

I'm wondering if it would be possible to have an entrylevel workshop here at PF on groups.

I mean a collective teach-each-other tutorial-----no one person doing all the teaching but trading around.

I see Tom and Rutwig and Chroot have posted online resources
and also hardcopy books to buy.

The big question is------is there enough interest?
A secondary question is-----could we stand to type all the subscripts, superscripts, matrices, and greek letters? PF is a great medium for non-hierarchical learning. But the sheer typing of symbols and inability to draw pictures imposes some limits on what one can handle here.

So I am skeptical that a group theory tutorial or workshop would get anywhere.

But just to see how it might go----here is my proposal

Focus on the simplest most classical groups central to basic physics--dimensions 2, 3, 4.

Focus on things like SO(3) the special orthogonal group. ["special" just means det = 1 in this case, think of rotations]

And SU(2) the special unitary group----because of its relation to SO(3) and the pauli spinmatrices. among other things.

And SL(2,C) because of its relation to the Lorentz group.

It seems to me that the goal should be not to snow anybody or discourage anybody----not to show off or try to pull rank on people (as non-PF people sometimes do when discussing math)----but simply to go over the group theory that is most basic and do it in an entrylevel way.

This might not be possible---it might simply not work.

Also it might be tiresome to try to type in matrices---even like the three pauli spinmatrices which are about as simple as 2x2 matrices can get would be sort of tedious to type into PF-style posts.

Anyway I am broaching the idea. Reactions? Better ideas of how to do it?

Lonewolf

I like the sound of your idea, Marcus. I'd be interested once my exams are done. I like especially the sound of learning its applications to Physics. We get taught Group Theory, but only in the sense of pure maths.

marcus

If anyone knows good notes on the web that correspond to what Lonewolf is talking about (basic classical group theory with an eye to applications in physics) please post a link.

Lonewolf, this thread may possibly remain dormant until you are thru exams. Depends on how interested the others are. When you or anybody returns I will probably get a notice by email. but to be sure, send me a PM.

rutwig

The book by Cvitanovic is one of the links. There are many others, but it should be specified whether one is interested on discrete, continuous (non differentiable) or Lie groups, or even generalizations like Kac-Moody groups, supergroups, etc. Each of the topics is a world in itself.

marcus

Rutwig I do not know if you have any interest in LQG or follow it at all but, if you do, then you probably have noted that
a recent result of Olaf Dreyer seems to force a change in the group from SU(2) to SO(3)
Lubos Motl has what seems to be a clearsighted outsiders
perspective on this (not being especially an advocate of LQG)

Have you any comment on this----perhaps the change seems insignificant given that one is a 2-fold cover of the other---or does it have some interesting ramifications?

I will edit this to add a link to Lubos Motl's paper, though
I would not be surprised if you had already noticed it.

http://www.arxiv.org/abs/gr-qc/0212096

Kalimaa23

A good place for online textbooks is

http://www.math.gatech.edu/~cain/tex...linebooks.html

I printed out "Abstract Algebra, the basic graduate year" by Prof. Robert Ash, and it looks pretty good so far.

A workshop would be pretty nice, since I had already planned to study some algebra this summer anyway.
I got an introduction this semester, and although most of my fellow physics students hated the abstactness of it all, it grew on me. Seems like a fun game to play.

rutwig

Interesting ramifications should be searched for experimentally, but it is not at all insignificant that the adequate group is not the simply connected universal cover, but some projection of it. With respect to the covering, this would indicate that the system makes no distinction of the covering elements (as happens at the tangent space level), and probably this has some significant consequences.