# $GL(n,\mathbb R)/SL(n,\mathbb R)$ is isomophic to $\mathbb R^\times$

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 $G/S \cong R^{\times}$.

• Are you familiar with the determinant, in particular the fact that it is a homomorphism? – Alex Becker Apr 14 '13 at 20:15
• @AlexBecker, it would be very difficult for the OP to know what SL is otherwise... – Mariano Suárez-Álvarez Feb 1 '18 at 19:03

## 1 Answer

If you know that $\det:G\to \mathbb R^\times$ is a surjective homomorphism, then the exercise is easy. Observe that $\ker\det = S$ (since $1$ is the identity of $\mathbb R^\times$) and so by the first isomorphism theorem, $\mathbb R^\times\cong G/\ker \det = G/S$.

• do I have to show that det:$G \rightarrow R^{x}$ is a onto homomorphism or can I just assume? – Gamecocks99 Apr 14 '13 at 20:29
• @Gamecocks99 That depends on whether it has been shown/assumed in your class? – Alex Becker Apr 14 '13 at 20:30
• $$\forall\,r\in\Bbb R\;,\;\;\det\begin{pmatrix}r&0&0&\ldots&0\\0&1&0&\ldots&0\\\ldots&\ldots&\ldots& \ldots&\ldots\\0&0&\ldots&0&1\end{pmatrix}=r$$ – DonAntonio Apr 14 '13 at 21:02