Quotient Group of General Linear Group by Special Linear Group

Theorem

Let $\GL {n, \R}$ denote the general linear group of degree $n$ over $\R$.

Let $\SL {n, \R}$ denote the special linear group of degree $n$ over $\R$.

Then the quotient group $\GL {n, \R} / \SL {n, \R}$ is the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

Proof

Let $\det: \GL {n, \R} \to \struct {\R_{\ne 0}, \times}$ be the group homomorphism:

$\mathbf A \mapsto \map \det {\mathbf A}$

where $\map \det {\mathbf A}$ is the determinant of $\mathbf A$.

This is demonstrated to be a homomorphism in General Linear Group to Determinant is Homomorphism

From the the corollary to General Linear Group to Determinant is Homomorphism, the kernel of $\phi$ is $\SL {n, \R}$.

Thus from Kernel is Normal Subgroup of Domain, $\SL {n, \R}$ is normal in $\GL {n, \R}$.

From the First Isomorphism Theorem for Groups:

$\Img \phi \cong \GL {n, \R} / \SL {n, \R}$

By definition, the image of $\phi$ is the multiplicative group of real numbers.

Hence the result.

$\blacksquare$