Classifying Groups: Finite, Discrete, Continuous

[Herr Malus]

I have a question regarding terminology here. The assignment is somewhat as follows: "If you think any of the following is a group, classify it along the following lines: finite, infinite discrete, finite-dimensional continuous, infinite-dimensional continuous."

The definition of finite is obvious, but I haven't been able to find much of anything on the other three. Anyone have any good examples, or could at least point me in the correct direction?

 

[Dick]

This involves more than just group theory. Discrete is a topology term. The group of integers under addition is discrete in the usual topology. And if you are talking about dimension that only applies to vector spaces. And I'm guessing 'continous' means you are talking about manifolds. Do you have at least a rough idea what these mean?

 

[Herr Malus]

If discrete is a reference to the discrete metric for a discrete topology, then yes. So in that case I assume we want a group of isolated elements. In differentiating between finite dimensional and infinite dimensional, are we just looking for the order of the group? I.e. the number n such that g^n=e, where e is the identity element of the group? Finally, my knowledge of manifolds comes from a G.R. course, so would I be looking for some sort of connection to the Lie groups there?

 

[Dick]

If discrete is a reference to the discrete metric for a discrete topology, then yes. So in that case I assume we want a group of isolated elements. In differentiating between finite dimensional and infinite dimensional, are we just looking for the order of the group? I.e. the number n such that g^n=e, where e is the identity element of the group? Finally, my knowledge of manifolds comes from a G.R. course, so would I be looking for some sort of connection to the Lie groups there?

Dimension has to mean dimension as a vector space or manifold. Not as a group. What kind of groups are you trying to classify?

 

[Herr Malus]

Examples of the types of groups we're looking at are:
-"The set of Mobius transformations in the complex plane", where I assume the operation is composition.
-"The set {1,2,3,...,p-1} under multiplication modulo p, where p is a prime number."

 

[Dick]

Ok, what do you think about those two?

 

[Herr Malus]

Well, the first is in the complex plane, C, so I'd assume dimension is 2. For the second, I think I actually want order (so I picked a bad example, my apologies), in which case it seems that the order is p-1.

 

[Dick]

Well, the first is in the complex plane, C, so I'd assume dimension is 2. For the second, I think I actually want order (so I picked a bad example, my apologies), in which case it seems that the order is p-1.

The mobius transformations form a Lie group. So it's a continuous group. The dimension of the group is going to be the dimension of the Lie group manifold, not the dimension of C. A mobius transformation (az+b)/(cz+d) is specified by the parameters a, b, c and d. So you can certainly say it's finite dimensional. And sure, the second one is finite.