Are there any sorts of multiplicative group structures we can put on set $\mathbb{R}^n$? By multiplicative, I mean that it should be compatible with the natural scalar multiplication on $\mathbb{R}^n$; that is, for any $x,y \in \mathbb{R}^n$ and $c \in \mathbb{R}$, we should have $ c(x \star y) = cx \star y$
For $n > 2$, can such structures be abelian? The only example I'm familiar with is $n = 4$, in which we have the quaternions; this is clearly non-abelian.