I have the following exercise: Let $H$ be a finite subgroup of $Isom(\mathbb{R}^n)$, i.e. the group of all isometries of $\mathbb{R}^n$. Prove that there exists a set $X\subset \mathbb{R}^n$ such that $H$ is the symmetry group of $X$.
I think that we can start with a point $x\in \mathbb{R}^n$ and consider the orbit of this point under the action of $H$, however this orbit set might be too big. So I am thinking about adding more points to this orbit so that we can reduce its symmetry group, however it is not so clear...
Can someone help me? Thanks a lot!