I'm asked to show that $GL_2(\mathbb{R})/S$ is isomorphic to $Z_2$ where
$GL_2(\mathbb{R})=$$\left\{A= \begin{pmatrix} a & b\\ c & d \end{pmatrix}, |a,b,c,d \in \mathbb{R} , det A \neq 0 \right\}$
$S=$$\left\{A \in GL_2(\mathbb{R}) |a,b,c,d \in \mathbb{R} , det A > 0 \right\}$
$Z_2$ is the additive group of residue class module 2
I have proved S is a normal subgroup of $GL_2(\mathbb{R})$ but can't find a mapping to prove $GL_2(\mathbb{R})/S$ is isomorphic to $Z_2$