Find all subgroups of $\mathrm{GL}(2,\mathbf{R})$ of index 2.

Question

Question: We want to find all subgroups of $\mathrm{GL}(2,\mathbf{R})$ of index $2$.

Here is my first attempt: We already know that for a group $G$, if a subgroup $H$ satisfies $(G\colon H)=2$, then $H$ is a normal subgroup of $G$. So we first try to construct some normal subgroups of $\mathrm{GL}(2,\mathbf{R})$. Note that $$S\,\colon\!=\left\{A\in \mathrm{GL}(2,\mathbf{R})\,|\,\det(A)>0\right\}$$ is a normal subgroup of $\mathrm{GL}(2,\mathbf{R})$. And $\left|\mathrm{GL}(2,\mathbf{R})/S\right|=2$. So $S$ is one subgroup meeting our requirement.

But I am not sure whether there are other subgroups of $\mathrm{GL}(2,\mathbf{R})$ have index $2$.

Answer 1

As the comment given by @diracdeltafunk suggests, since $H$ is a subgroup of index 2, it suffices to classify all group homomorphism $f:\mathrm{GL}_2(\mathbb{R})\to \mathbb{Z}/2\mathbb{Z}$. Since $\mathbb{Z}/2\mathbb{Z}$ is commutative and the commutator of $\mathrm{GL}_2$ is $\mathrm{SL}_2$,see Commutator Group of $\operatorname{GL}_2(\mathbb{R})$ is $\operatorname{SL}_2(\mathbb{R})$

the homomorphism $f$ factors through $\mathbb{R}^\times$. It should be very easy to classify all group homomorphism $\overline {f}: \mathbb{R}^\times\to \mathbb{Z}/2\mathbb{Z}$.