# Prove that Sym(F) is a subgroup of O2(R)

1. Mar 5, 2013

### kasperrepsak

1. The problem statement, all variables and given/known data
$\textbf{26.}$ Let $F \subset \textbf{R^2}$ be a non-empty subset of $\textbf{R^2}$ that is bounded. Prove that after chosing appropriate coordinates $Sym(F)$ is a subgroup of $O_2(\textbf{R}).$

2. Relevant equations
The hints given are:
Prove there is an a$\in \textbf{R^2}$ so that for all φ in $Sym(F)$, φ(a)=a, and use that a as the origin.
Show that the mapping L: $Sym(F)$--> $O_2(\textbf{R})$ is injective, that all non trivial φ in $Sym(F)^+$ have a unique fixed point $a_φ$, and that $Sym(F)^+$ is commutative.

3. The attempt at a solution
I have recently started Abstract Algebra and this problem is supposably very difficult to prove. I know that since it is a bounded set there cant be a translational-symmetry like there could be for an infinite line. I'm not sure yet how to prove that, and if it is needed to prove the problem. Any help would be very welcome.

Last edited: Mar 5, 2013
2. Mar 6, 2013

### kasperrepsak

Could someone please help me on my way? It is a homework assignment for tomorrow : D.