Let $G$ be the group of all $n\times n$ matrices with real entries that are invertible. The operations of the group are matrix multiplication, matrix inversion, and identity matrix. Let $S$ be the set of $n\times n$ matrices with real entries and with determinant $1$. Let $R^\times$ denote the group of nonzero numbers with the operations of multiplication, multiplicative inverse, and $1$.
Prove that $G/S \cong R^{\times}$.