abstract algebra - Subgroup of multiplicative group of nonzero real numbers $Bbb R^*$ with index $2$

Let $\mathbb{R}^*$ denote the multiplicative group of nonzero real numbers. Is there a subgroup of $\mathbb{R}^*$ with index $2$?

Answer

Show that the signum function $\operatorname{sgn}$ is a homomorphism from $\Bbb R^*$ onto the multiplicative group $\{-1, 1\}$. The kernel of this homomorphism has index $2$.