# Questions tagged [linear-groups]

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### Abelianization of $\mathrm{GL}_n(\mathbb{Z})$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$What is the abelianization of $\GL_n(\mathbb{Z})$? I know the abelianzation of $\GL_n(\mathbb{F})$ where $\mathbb{F}$ is a field and the ...
• 721
1 vote
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### Infinite closed subgroup of ${\rm SL}_{n}(\mathbb{F}_{p}[[T]])$ with full residual image

Let $\mathbb{F}_{p}$ be a finite field of order $p$, $\mathbb{Z}_p$ be the ring of $p$-adic integers and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. For $p\geq 5$, ...
• 561
1 vote
81 views

### Non-Noetherian closed subgroups of ${\rm GL}_{n}(\mathbb{F}_{q}[[T]])$

Let $\mathbb{F}_{q}$ be a finite field of order $q$, and $\mathbb{F}_{q}[[T]]$ be the ring of formal power series over $\mathbb{F}_{q}$. We say that a profinite group $G$ is Noetherian if any closed ...
• 561
336 views

### Proving certain triangle groups are infinite

[Cross-posted from MSE] Consider the Von Dyck group $$G = \langle x,y\mid x^a=y^b=(xy)^c=1\rangle$$ where $a,b,c\ge3$. Because $G$ is infinite and residually finite, it has an infinite family of ...
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• 369
1 vote
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### Conjugacy classes and normal form of $O_n$ and $U_n$

I'm interested in characterizing conjugacy classes inside $O_n$ and $U_n$ over local fields of positive characteristic ($\neq 2$). I need this for my research on representation theory of these groups. ...
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• 275
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### Finite simple subgroups of $\mathrm{GL}_n(\mathbb{C})$

Let $n$ be a fixed positive integer. Is it true that there are only finitely many isomorphism classes of finite nonabelian simple subgroups of $\mathrm{GL}_n(\mathbb{C})$? I'm especially interested ...
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### Does every linear group admit a subgroup of dimension 1?

Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup? I'm pretty much sure this is true in ...
1 vote
275 views

### centralizer of a n-cyclic permutation matrix over F_2 in GL(n,2)

This is a continuation of this question, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit ...
• 13.9k
426 views

### centralizer of the order 2^k cyclic permutation matrix over F_2

Let $C$ be the $2^k\times 2^k$-permutation matrix over $\mathbb{F}_2$ of the $2^k$-cycle. We needed to know the structure of its centralizer in $\mathrm{GL}_{2^k}(\mathbb{F}_2)$, and we computed it - ...
• 13.9k
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### Restricting the composition factors of subgroups of GL_m(Z/nZ)

For a positive integer $m$, let $\mathcal{A}(m)$ be the set of all integers $k \geq 5$ such that: there is a positive integer $n$ and a subgroup $G \subset \operatorname{GL}_m(\mathbb{Z}/n\mathbb{Z})$ ...
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### Representation of surface group

Is there a faithful representation of a surface group of genus $>2$ into $GL(n,\mathbb{C})$ for some $n$ for which, for each conjugacy class of each embedded loop in the fundamental group, the ...
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### When is a Baumslag-Solitar group linear?

The Baumslag-Solitar group $BS(m,n)$ is given by the group presentation $BS(m,n)=(a,b|ba^{m}b^{-1}=a^{n})$. When does it embed into a linear group? Thanks!
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### A conjecture on Zassenhaus groups

In a 1995 paper by Ali Nesin, "Permutation groups of finite Morley rank", the following conjecture is mentioned: Conjecture 5 An infinite Zassenhaus group $G$ of finite Morley rank is isomorphic ...
• 215
Let $p$ be a prime number and $n,l$ be natural numbers. I'm interested in the conditions under which the general linear groups of degree $n$ over the following two length $l$ finite discrete valuation ...