All Questions
16
questions
1
vote
0
answers
154
views
Centraliser of a finite group
Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$.
We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
- 501
4
votes
0
answers
133
views
Derived subgroup of rational points vs. rational points of derived subgroups
Let $G$ be a connected split reductive group over a field $k$. In general, we have an inclusion
$$
f: [G(k), G(k)] \rightarrow [G,G](k).
$$
If $k$ is not algebraically closed, $f$ is not necessarily ...
- 2,711
5
votes
1
answer
343
views
Characters of tori in finite reductive group
Let $G$ be a connected split reductive group over a finite field $k$. Suppose $G$ has connected centre. Let $T$ be a maximal split torus with Weyl group $W$. Note that $W$ acts on the finite group $T(...
- 2,711
2
votes
0
answers
153
views
Reference request - obtaining finite simple groups from algebraic groups
I'm looking for references for the following statements, which I believe are true:
Let $G$ be a simply connected simple linear algebraic group over a finite field $k$ of cardinality $q\ge 4$. Let $Z\...
- 7,313
1
vote
1
answer
267
views
Twisted forms of $\mathrm{SL}(2,q)$
$\DeclareMathOperator\SL{SL}$Let $q = p^r$ be a prime power. Let $H$ denote the subgroup of $\SL(2,\overline{\mathbb{F}}_q)$ consisting of matrices of the form $\begin{pmatrix}a & b\\ b^q & a^...
- 7,313
7
votes
1
answer
743
views
Number of conjugacy classes of finite reductive groups
Let $G$ be a connected reductive group over $\mathbb{Z}$. Let $c_{G(\mathbb{F}_q)}$ be the number of conjugacy classes of $G(\mathbb{F}_q)$.
Question: Is it true that $c_{G(\mathbb{F}_q)}$ is a quasi-...
- 2,711
2
votes
1
answer
1k
views
Viewing a finite group as a group scheme
I've read books which have this statement, without explanation : 'every finite group is an algebraic group'. I'm trying to understand what exactly they mean. The definition I have in my mind of a ...
- 896
25
votes
0
answers
985
views
Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group
Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$?
I have no real ...
- 153k
2
votes
1
answer
1k
views
Classification of finite group schemes over a field
What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field.
Is there a full ...
- 21
6
votes
2
answers
413
views
How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?
Of course the general answer to the question in the title is: not very simple.
I could not think of a better title, so let me explain my question in more detail.
I have a number field $E/\mathbb{Q}$, ...
- 5,454
1
vote
1
answer
214
views
Decomposing quasi-finite separated group schemes
Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...
5
votes
2
answers
836
views
Quotient of a rational variety by a finite group
Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$,
$$G\times(X\times...\times X)\...
- 8,922
11
votes
2
answers
1k
views
Finite subgroups of $PGL(3,K)$
It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
- 125
11
votes
2
answers
949
views
Spherical building of an exceptional group of Lie type
I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
- 805
5
votes
2
answers
667
views
Finite group scheme acting on a scheme such that there is an orbit NOT contained in an open affine.
In Mumfords book on abelian varieties there is a theorem (on page 111) whose hypothesis is "Let G be a finite group scheme acting on a scheme X such that the orbit of any point is contained in an ...
- 467