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

Recall that $SL_n(\mathbb{Z})$ is the special linear group, $n\geq 2$, and let $q\geq 2$ be any integer. We have a natural quotient map $$\pi: SL_n(\mathbb{Z})\to SL_n(\mathbb{Z}/q).$$ I remember that this map is surjective (is it correct?). It seems the Chinese Remainder Theorem might be helpful, but I forgot how to prove it. Can anyone give […]

Let $G$ be a transitive, non-regular finite permutation group such that each non-trivial element fixing some point fixes exactly two points. Suppose that $G \cong PSL(2, q), q > 5$ and $H = G_{\alpha}$ for some point $\alpha \in \Omega$. Assume $q = p^n$. i) If $p \ne 2$ and $|H|$ is even, then $H$ […]

I need to find a finite generating set for $Gl(n,\mathbb{Z})$. I heard somewhere once that this group is generated by the elementary matrices – of course, if I’m going to prove that $GL(n,\mathbb{Z})$ has a finite generating set, I would need to prove that any matrix $M\in GL(n,\mathbb{Z})$ can be generated by only finitely many […]

Background There’re some nomenclatures from Michael Artin’s Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. $SU_2$ is a special unitary group, i.e, the group of complex $2\times 2$ matrices of the form \begin{bmatrix} a&b\\ -\bar b&\bar a \end{bmatrix} with $\bar aa+\bar bb=1$. Rewrite $a=x_0+x_1i,b=x_2+x_3i$, there’s an obvious 1-1 […]

Consider the transpose inverse automorphism on $GL_n(\mathbb F)$ where $n\geq2$ and $|\mathbb F|>2$. (i.e. $\mathbb F$ is a field, possibly infinite, with three or more elements). I want to show this automorphism is not inner. I was told to consider $\det(BAB^{-1}) = \det(A)$ and $\det(\,^TA^{-1})=\det(A)^{-1}$ and derive a contradiction. However, in some fields (where the […]

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