The linear-groups tag has no wiki summary.

**3**

votes

**2**answers

146 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

471 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

122 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 ...

**4**

votes

**0**answers

220 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

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

**5**

votes

**1**answer

209 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

486 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

139 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

187 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 ...

**33**

votes

**2**answers

971 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 ...

**10**

votes

**3**answers

858 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

289 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

248 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 ...