TITLE: Determine all subgroups of $\mathbb{R}^*$ (nonzero reals under multiplication) of index $2$.
QUESTION [8 upvotes]: Determine all subgroups of $\mathbb{R}^*$ (nonzero reals under multiplication)
of index $2$.

how can I able to solve this problem

REPLY [3 votes]: First, notice that $|\cdot | : x \mapsto | x|$ and $\operatorname{sign} : x \mapsto \left\{ \begin{array}{cl} 1 & \text{if} \ x>0 \\ -1 & \text{if} \ x<0 \end{array} \right.$ are surjective homomorphisms from $\mathbb{R}^{\times}$ to $\mathbb{R}_{>0}$ and $\mathbb{Z}_2$ respectively, with $\text{ker}(\operatorname{sign})=\mathbb{R}_{>0}$. In particular, $\mathbb{R}^{\times} \simeq \mathbb{Z}_2 \times \mathbb{R}_{>0}$ (for the isomorphism, take $x \mapsto (\operatorname{sign}(x),|x|)$) and $\mathbb{R}_{>0}$ is of index two in $\mathbb{R}^{\times}$.
Let $H$ be a subgroup of finite index in $\mathbb{R}^{\times}$; then $|H|$ is of finite index in $\mathbb{R}_{>0}$. But $\mathbb{R}_{>0}$ is an infinite divisible abelian group, so $|H|=\mathbb{R}_{>0}$ hence $\mathbb{R}_{>0} \subset H$. 
You deduce that $[\mathbb{R}^{\times}: H] \leq [\mathbb{R}^{\times}: \mathbb{R}_{>0}]=2$, that is $H= \mathbb{R}^{\times}$ or $H=\mathbb{R}_{>0}$.