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

[kasperrepsak]

Homework Statement

$\textbf{26.}$ 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 $O_2(\textbf{R}).$

Homework 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.

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 can't 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.

 

[kasperrepsak]

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