Consider the problem of classifying the finite groups of isometries of R^n. --For n=2 it is cyclic and dihedral groups. --For n=3 they are well known, probably from Kepler and are related to ade-classification. --For n=4 we can get them by taking the universal cover of SO(4) which is isomorphic to SU2 x SU2, though I do not know where the classification is available$\mathbb{R}^n$.
- For $n=2$ it is cyclic and dihedral groups.
- For $n=3$ they are well known, probably from Kepler and are related to ade-classification.
- For $n=4$ we can get them by taking the universal cover of $\mathrm{SO}(4)$ which is isomorphic to $\mathrm{SU}(2) \times \mathrm{SU}(2)$, though I do not know where the classification is available.
But my main question is for dimension n=5 and above$n\geq 5$. Does anybody knows the state of the art? A A reference would be most helpful. Note
Note that the finite subgroups of GLn(Z)$\mathrm{GL}_n(\mathbb{Z})$ are classified for n<=10$n\leq 10$.
Mathieu
