# Questions tagged [general-linear-group]

The general linear group of order $n$, $GL_n(\mathbb{F})$, over a field $\mathbb{F}$ (usually $\mathbb{R}$ or $\mathbb{C}$) is the group of $n\times n$ invertible matrices over $\mathbb{F}$. The operation is the usual matrix multiplicatoin.

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### The subgroup $B_n \leq \mathrm{GL}_n(F)$ of upper triangular matrices is solvable (Is my proof correct?)

I do my group theory homework again and after some thought I came up with the following proof. It would be nice, if someone could tell me if this is correct. I took some inspiration from this previous ...
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### Does every spherical sector of a circle in $\mathrm{M}_n(\mathbb{C})$ contain an invertible matrix?

Some context : (Notations : For the rest of the post, I will identify $\mathrm{M}_n(\mathbb{C})$ with $\mathbb{C}^{n^2}$, and I will note $\mathcal{S}^{n^2-1}$ the unit sphere of $\mathbb{C}^{n^2}$) I ...
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### References for Maltsev's Theorem on $GL_n(F) \equiv GL_m(K)$ iff $K \equiv F$ and $m=n$.

I have recently found Maltsev's theorem: for $F$ and $K$ algebraically closed fields, $GL_n(F) \equiv GL_m(K)$ if and only if $K \equiv F$ and $m=n$ (thanks to this question: https://mathoverflow.net/...
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### Does there exist an element of order $4$ in $GL_2(\mathbb{Z})/GL_2(\mathbb{Z})'$?

For a group $G$, let $G'$ denote the commutator of the group $G$, and if $H \leq G$ the left cosets will be denoted as $gH$. Now, I understand the fact that $[SL_2(\mathbb{Z}):GL_2(\mathbb{Z})'] = 2.$ ...
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