# Abelianization of general linear group?

I am asking purely out of interest: What the abelianization of general linear group $GL(n,\mathbb{R})$?

The derived group of ${\rm GL}(n,K)$ is ${\rm SL}(n,K)$ for all $n \ge 1$ and all fields $K$, so the abelianization of ${\rm GL}(n,K)$ is the multiplicative group of the field, $(K \setminus \{0\},\times)$.
• Except when $n=2$ and $K = \mathbb{F}_2$. In that case, $GL(2,\mathbb{F}_2)$ is isomorphic to $\Sigma_3$ and its abelianization in $\mathbb{Z}/2$. – Goa'uld Jun 10 '16 at 23:54
• I know this is an old post - but it's interesting to point out that except in the exceptional case, the determinant morphism $GL(n, K) \to K^*$ is a universal abelianization morphism. – Daniel Schepler May 11 '17 at 22:46