Proving there is a fixed point in a discrete group of rotations

[SNOOTCHIEBOOCHEE]

Homework Statement

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'

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 t_arho_theta.

and phi: G----> O

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

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.

 

[SNOOTCHIEBOOCHEE]

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

but i still don't know how to do this.

 

[SNOOTCHIEBOOCHEE]

anybody?

 

[HallsofIvy]

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

 

[SNOOTCHIEBOOCHEE]

the image of G in O.

Oh snap p=(0,0)