Insight about compact groups I'm quite familiar with the general notion of compactness in math but I have some troubles with its extension to group theory. I'm not talking about definitions or theorems: I would like to have some insight/intuition about why some groups are compact and why others aren't, what can we do with compact groups that can't be done with non-compact ones? Why is the Lorentz group in Physics not compact for instance? How can we spot them? Some examples would be great!
Thanks!!
 A: You are putting together two structures: that of a group and that of a topological space, in order for the two to be related you will want the group operations to be continue. Such a thing is called a topological group. Since you can view a topological group as a topological space with added structure, there is no why some groups are compact and some other not, you simply have started with compact/non compact top spaces. For example $\mathbb{R}$ and $S^1$ can be made into topological groups.
There may very well be theorems about topological conditions for a space to admit a group structure, maybe to the point of saying that it is "easier" to have a group structure on a non-compact topological space, but I do not know. It is well known, but hard to prove, that the only spheres admitting a group structure are those of dimension 1,3,7 (trivially also in dimension 0 where you get $\mathbb{Z}_2$) and this has to do with $R,C,H$ being the only division algebras.
I believe I read somewhere that any topological group can be given a unique (?) smooth structure (but double check this) with respect to which the group operations are smooth. Such a group is called a Lie group. An easy necessary condition for a space to be a Lie group is to have a trivial tangent bundle, this follows from the identification between the tangent space at the identity and the left invariant vector fields. This rules out many smooth manifolds from being Lie groups.
The main thing we can do with compact groups but not with non-compact ones is integrate on them. The possibility of integrating allows one to extend many results about finite groups to compact groups. Moreover representation theory for compact groups is much easier than for non-compact ones. In fact I think that the problem of classifying (finite dimensional ?) representations of compact Lie groups is completely solved as it can be reduced to that of classifying the representations of compact semisimple Lie groups which come into 4 infinite families and 5 exceptional groups.
Intuitively the reason why the Lorentz group is non-compact is that the parameter of a boost can take arbitrary large values, in fact as a top space the Lorentz group is homeomorphic to $\mathbb{R}^3\times SO(3)$. 
