Let $G$ be a finite group of odd order. Suppose that $G$ has a *real* 4-dimensional faithful representation. Is it true that $G$ should be abelian in this case? 



Real representation of group of odd order
We can assume $G\leq O(4)$ wlog. Because $|G|$ is odd, we have in fact $G\leq SO(4)$ ($g\mapsto g^2$ is a bijection and hence every $g$ must have positive determinant) and a monomorphism $G\to PSO(4) \cong SO(3)\times SO(3)$. Now lets look at the two projections $\pi\_1(G), \pi\_2(G)\leq SO(3)$. We consult the [list of finite order subgroups of $SO(3)$](http://groupprops.subwiki.org/wiki/Classification_of_finite_subgroups_of_SO%283,R%29) and find that the only odd-order subgroups of $SO(3)$ are the cyclic ones. Therefore $G\leq \pi\_1(G)\times\pi\_2(G)$ is abelian.


