Identify the quotient group $G/H$

Question

Let $G=GL_n(\mathbb{R})$ and let $H$ be the normal subgroup of $G$ of all matrices with positive determinant. Identify the group $G/H$

I know the trick for this is to define a homorphism from $G$ to something such that the kernal is $H$, then we can apply the isomorphism theorem. But I'm not able to identify the mapping.

Answer 1

Let $h:GL_n(\mathbb R)\to\mathbb Z_2$ be defined by $h(A)=\text{sgn}(\det(A))$ (where $\text{sgn}$ gives the sign of a number). This is a homomorphism because $\det(AB)=\det(A)\det(B)$ and $\text{sgn}(xy)=\text{sgn}(x)\text{sgn}(y)$. The kernel is exactly $H$, so $G/H\cong\mathbb Z_2$ by the first isomorphism theorem.