The linear-groups tag has no usage guidance.

**1**

vote

**0**answers

75 views

### Profinite rank of Fuchsian groups

Let $G$ be a Fuchsian group whose profinite completion is finitely generated. Must $G$ be finitely generated?
A Fuchsian group is a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ or ...

**2**

votes

**1**answer

91 views

### When is the image of the adjoint representation of a real algebraic group Zariski closed?

Let $\operatorname{Ad}:\operatorname{SL}_n(\mathbb{R}) \to \operatorname{GL}(\mathfrak{sl}_n(\mathbb{R}))$ be the adjoint representation (i.e. $\operatorname{Ad}(g)X=gXg^{-1}$) of $SL_n(\mathbb{R})$. ...

**2**

votes

**1**answer

74 views

### The maximal possible rank of a subgroup of a product of special linear groups

In this question I ask for a generalization of What is the maximal possible rank of a subgroup of a special linear group mod a prime?
Let $p_1, \dots, p_r$ be $r$ distinct odd primes.
Set $$G = ...

**7**

votes

**1**answer

97 views

### What is the maximal possible rank of a subgroup of a special linear group mod a prime?

Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$.
What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$?
Here we denote by $d(G)$ the smallest ...

**5**

votes

**1**answer

162 views

### Maximal subgroups of special linear groups over finite fields

Let $p$ be a prime number, and denote by $\mathbb{F}_p$ the field with $p$ elements.
Is there a classification of the maximal subgroups of $G = \mathrm{SL}_3(\mathbb{F}_p)$ ?
I am interested in ...

**3**

votes

**1**answer

158 views

### Is there a bound on the rank of finite index subgroup of SL_3(Z)?

Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}_3(\mathbb{Z})$ has a generating set of size $N$?

**13**

votes

**0**answers

301 views

### Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...

**1**

vote

**0**answers

137 views

### Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...

**13**

votes

**2**answers

425 views

### Minimal number of generators for $GL(n,\mathbb{Z})$

$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = ...

**10**

votes

**1**answer

376 views

### The Mordell and Bogomolov problems in linear groups

Many things in the arithmetic of abelian varieties have counterparts not only in linear tori, but also for semisimple linear groups. Two examples are the Tamagawa number and the conjectured finiteness ...

**2**

votes

**1**answer

316 views

### Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated?

Let $n \in \mathbb{N}$, $K$ a finite field. Denote by $K[[x]]$ the (profinite) ring of formal power series over $K$. Note that $\text{GL}_n(K[[x]])$ is a profinite group.
Is every closed subgroup of ...

**0**

votes

**0**answers

166 views

### classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by
$$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto ...

**6**

votes

**2**answers

308 views

### Maximal order of finite subgroups of $GL(n,Z)$

I am interested in the finite subgroups of $GL(n,Z)$ of maximal order.
Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...

**3**

votes

**2**answers

170 views

### differences between character distributions of supercuspidal representations and others

Let $G$ be a $p$-adic linear reductive group. For an irreducible admissible smooth representation $\pi$ of $G$, there is a distribution $\Theta(\pi)$, called the character distribution, attached to ...

**5**

votes

**3**answers

530 views

### 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

**1**answer

167 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 ...

**5**

votes

**1**answer

292 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 - ...

**4**

votes

**2**answers

270 views

### 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 $G \subset \operatorname{GL}_m(\mathbb{Z}/n\mathbb{Z})$ ...

**6**

votes

**1**answer

253 views

### 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 ...

**5**

votes

**2**answers

805 views

### 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!

**4**

votes

**0**answers

172 views

### 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 ...

**6**

votes

**1**answer

229 views

### What are the necessary and sufficient conditions for GL(n,Z/p^lZ) to be isomorphic to GL(n,F_p[t]/t^l)?

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 ...

**38**

votes

**2**answers

1k views

### Isomorphic general linear groups implies isomorphic fields?

Suppose $n > 1$ is a natural number. Suppose $K$ and $L$ are fields such that the general linear groups of degree $n$ over them are isomorphic, i.e., $GL(n,K) \cong GL(n,L)$ as groups. Is it ...

**11**

votes

**0**answers

294 views

### distribution of Young diagrams

Consider $\Lambda^p(C^n\otimes C^n)=\oplus_{\pi}S_{\pi}C^n\otimes S_{\pi'}C^n$ as
a $GL_n\times GL_n$-module. This space has dimension $\binom {n^2}p$. I would
like any information on the shapes of ...

**10**

votes

**3**answers

1k views

### Sylow subgroups of projective general linear groups

What is known about the Sylow 2-subgroups of $\rm{PGL}_n(\mathbb{F}_q)$, where $q$ is any prime power? For example, according to Theorem 7.9 in Isaacs's Character Theory, these Sylow 2-subgroups ...

**8**

votes

**1**answer

296 views

### On which space does $GL_n(F_p[X])$ act nicely?

The group $GL_n(\mathbb{Z})$ acts properly and isometrically on the space of homothety classes of scalar products on $\mathbb{R}^n$. This is a Riemannian manifold of nonpositive sectional curvature.
...

**2**

votes

**2**answers

273 views

### Existence of proper invariant subset in an irreducible action

Let $G<\rm{GL}_n(\mathbb{k})$ be a linear group, where $\mathbb{k}$ is an algebraically closed field. Assume that the linear action of $G$ on $\mathbb{k}^n$ is strongly-irreducible (i.e. there are ...

**6**

votes

**3**answers

1k views

### A free subgroup of GL(2,Z)?

Is the subgroup of $GL(2,\mathbb Z)$ generated by the matrices
$$ \left( \begin{array}{cc}
1 & 1 \\\
1 & 0 \end{array} \right) \ \ \text{and} \ \
\left( \begin{array}{cc}
2 & 1 \\\
1 ...