Understanding Group Theory

dpa

Hi Everyone,

I am kind of looking some online text to understand Lie Algebra, Group Theory and so forth.
I usually need application (everyday/science context how it is used) and intuition more than mere mathematical definition to understand topics. So I need some text that gives very deep understanding of the text.

And I am not sure whether that topic falls under Linear and Abstract Algebra or under some other thread.

Thank You

Sincerely
DPA

chiro

I don't know about text books, but I think I can offer one way to see how group theory is useful.

Consider a chess game. In chess we have rules for every chess piece in terms of moving a piece whether its just a movement or whether it involves moving your piece to steal an opponents piece.

Now consider the fact that you can always 'undo' every action that you carried out: in other words, if you gave me the list of every move you and your opponent gave for the entire game as well as the final state of the game: I could reverse the whole process all the way back to the original setup of the board.

If you think in terms of a symmetry, you can see that with a group, you can understand a particular kind of process symmetry in this chess example: one important property of groups is that they are reversible and thus have a form of symmetry in this context.

Now consider that instead of just chess pieces, we want to take this idea and consider all kinds of transformations. The transformation may involve rotating a shape or something along the lines of solving a rubix cube. It might also have to do with number theory, which is what has happened with crytography: because public-key cryptography can be seen in a group-theoretic context (you must be able to get back the original message from the encrypted message for cryptography to be useful), then by finding things out about groups, you also indirectly learn something about cryptographic algorithms.

This idea of reversibility, or inversibility, or symmetry or whatever you want to call it is an important way to study systems. Also if something really is a group, then it means that because of this symmetry you can create a process and undo it: I know I've said this many times but realize that many things in mathematics do not have this property, but if something is a group, then it must which means we can look at the consequences of something having this property.

The key word is consequence: mathematics (a large part of it at least) is concerned with taking an idea that seems useful and to exploring what the consequences of that idea has on particular attributes: in other words, we are given this object that has all these properties (inverses, identities, associativity and closure) and then we say "well what does this mean for these kinds of structures in the context of _____"

dpa

thank you chiro for the long post.

dpa

any other links to online sources would be helpful too

chiro

Sorry I don't know many online sources. The above post was just based on my thoughts and reflections over the years. I'm sure you will find something interesting if you add specific terms to your search which I don't know: just type in the first thing that comes in your head and take it from there :)

dpa

thank you.

dpa

could anyone else extend it to string theory, context how it is used there. I would else be asking too much from chiro.

mathwonk

there are some free sets of class notes on this page:

http://www.math.uga.edu/~roy/

Stephen Tashi

I don't know anything about string theory, but, as to group theory, Lie groups etc. , you can't expect to get a useful knowledge of these subjects without some ability and experience in doing formal mathematics.

You haven't explained your background. It isn't clear whether you have realistic goals.

chiro

If you want to understand groups in physics then just get a book that talks about this specifically. I did a very quick google search and got this:

http://www.amazon.com/Group-Theory-P.../dp/0521558859

There is bound to be other similar books like this that cover groups in physics or physics in a group theoretical perspective.