Homework Help: Proving there is a fixed point in a discrete group of rotations

1. Nov 23, 2008

SNOOTCHIEBOOCHEE

1. The problem statement, all variables and given/known data

Let G be a discrete group in which every element is orientation-preserving. Prove that the point group G' is a cyclic group of rotations and that there is a point p in the plane such that the set of group elements which fix p is isomorphic to G'

3. The attempt at a solution

Ok since every element of the group is orientation-preserving, we know that there are only translations and rotations. so every element of G can be written as tarhotheta.

and phi: G----> O

So the point group G' = phi(G) = rhotheta

Im guessing i have to use the fixed point theorem for the second part, but i have no clue how to do that.

Any help is appreciated.

2. Nov 23, 2008

SNOOTCHIEBOOCHEE

we need to find a point p such that all g(p)=p for all g in G.

but i still dont know how to do this.

3. Nov 24, 2008

SNOOTCHIEBOOCHEE

anybody?

4. Nov 24, 2008

HallsofIvy

What is the definition of "the point group G' "?

5. Nov 24, 2008

SNOOTCHIEBOOCHEE

the image of G in O.

Oh snap p=(0,0)

Last edited: Nov 24, 2008