# Tagged Questions

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67 views

### Symmetric spaces which are compact modulo the unipotent radical are compact

Is the following true?
Let $X = G/H$ be a symmetric space of a reductive group over a p-adic field $F$.
Let $X^0$ be an open orbit w.r.t. the action of the minimal parabolic $B$ of $G$.
Let $U$ be ...

**6**

votes

**1**answer

205 views

### Generalization of Frobenius groups

Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.
In other words, if in a ...

**4**

votes

**1**answer

166 views

### Automorphisms of SO_n(k,f)

Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let
$$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$
and
...

**3**

votes

**2**answers

146 views

### Elementary abelian $p$-subgroups of maximal rank in finite groups of Lie type

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a reductive group defined over $\mathbb{F}_p$. For any $d\in\mathbb{Z}^+$, let $C_d(G)$ be the set of conjugacy ...

**3**

votes

**1**answer

360 views

### Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...

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vote

**4**answers

233 views

### About structure of parabolic subgroups of finite classical algebraic groups

Dear Members of Mathoverflow,
I am interested about a Fact (if it is right) of the structure of parabolic subgroups of finite classical algebraic groups:
Let G be a classical algebraic group over ...

**3**

votes

**2**answers

228 views

### Suslin's Stability Theorem for Chevalley Groups

I am looking for a version of Suslin's Stability Theorem for Chevalley groups.
The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary ...

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votes

**0**answers

72 views

### radical unipotent of a parahoric

Let $G$ a split connected reductive group over $\mathbb{C}$. $F=\mathbb{C}((t))$ and $\mathcal{O}$ the ring of integers.
Let $B$ a Borel subgroup and $I$ the corresponding Iwahori.
Let ...

**3**

votes

**1**answer

109 views

### Non-abelian group from affine hermitian curve

I was playing with the Hermitian curve $y^q + y = x^{q+1}$ over the field $GF(q^2)$ and chanced upon the following (non Abelian) group law on the points of the affine curve:
$(a,b) * (c,d) = ...

**2**

votes

**1**answer

158 views

### Which subgroups of a finite reflection group have distingushed coset representatives?

Let $W$ be a finite reflection group with length function $l$ and let $I$ be a set of simple reflections that generate $W$. Let $\phi$ be an automorphism of $W$ permuting $I$. Consider the orbits of ...

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votes

**1**answer

161 views

### When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$.
Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...

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votes

**2**answers

226 views

### Root system automorphisms as inner automorphisms of extended Chevalley group

For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...

**4**

votes

**1**answer

251 views

### Zariski dense subgroups of linear algebraic groups

The theorem of Matthews, Vaserstein and Weisfeiler asserts that if $G$ is a simply connected absolutely almost simple groups over $Q$ and $\Gamma$ a finitely generated subgroup of $G(Q)$ which is ...

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**2**answers

211 views

### Centralizer of a subtorus in a reductive group is Levi?

Questions a bit similar to this one have already appeared I think on the forum but I couldn't find the answer to my question using those answers. I must say from the beginning that my knowledge of ...

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**0**answers

229 views

### Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...

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**0**answers

93 views

### semisimple conjugacy classes over general bases

Let $k$ be an algebraically closed field, $G$ a connected reductive group, $T$ a maximal torus, $W$ the Weyl group and $\chi:G\rightarrow T/W$ the Steinberg morphism.
We know that if ...

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vote

**0**answers

139 views

### on the open bruhat cell

Let $G$ a connected reductive group and $S=U^{-}TU$ the open cell.
Do we have $G=\bigcup\limits_{g\in G}gSg^{-1}$?
And also if I assume that $G$ is adjoint and $\overline{G}$ is the de ...

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**0**answers

142 views

### Compute the discriminant for reductive groups

Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...

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votes

**1**answer

132 views

### One-parameter subgroups of symplectic group associated to roots

I'm having trouble sorting out some basic definitions concerning Chevalley groups. The groups I'm interested in are the simply connected groups of type $C_n$, so the groups $\text{Sp}_{2n}$. The ...

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**3**answers

566 views

### Spin group as an automorphism group

Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with ...

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**0**answers

161 views

### Certain central extensions of simply connected simple algebraic groups

An offbeat question involving Milnor's $K_2$ has come up recently. Start with an algebraically closed field $F$ (perhaps required to be of characteristic 0). Let $G$ be a connected, simply connected ...

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**2**answers

697 views

### Definition of “finite group of Lie type”?

The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from ...

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**1**answer

140 views

### Reference on elements of finite order in principal congruence subgroups of symplectic groups

We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with ...

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votes

**1**answer

242 views

### Counting conjugacy classes in simple groups of Lie type

Finite groups of Lie type include those obtained as rational points of a connected simple algebrraic group over a finite field $k = \mathbb{F}_q$ of characteristic $p$: these are split or quasi-split. ...

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**2**answers

340 views

### group generated by Coxeter elements

Let $G$ a connected semisimple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.
What can be said about $W'$, the subgroup of $W$ generated by the Coxeter elements of $W$?

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**2**answers

393 views

### Is there an almost-direct product decomposition for disconnected reductive algebraic groups?

$\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...

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**1**answer

374 views

### About isomorphism of $PGL(2)$ and $SO(3)$ [closed]

I need to prove that $PGL_2(\mathbb{R})\cong SO_3(\mathbb{R})$. Abstract considerations show that both can be identified with the group of projective motions of a conic curve. But maybe there is more ...

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425 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 ...

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**2**answers

402 views

### Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$

From a result Obtained by O. Schreier and B. L. van der Waerden [Math. Sem. Univ. Hamburg 6, 303- 322 (1928)], one can show that for two fields $\mathbb F$ and $\mathbb G$, and integers $n>m>2$, ...

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**1**answer

199 views

### Fixed points of group action

Let us consider the group $PGL(2,\mathbb{R})$ as the group of automorphisms of real projective line and $H\subset PGL(2,\mathbb{R})$ is a subgroup of prime order $> 2$. Is it true that there always ...

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**0**answers

217 views

### Reductive Lie Groups and Complexification

Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, ...

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**0**answers

136 views

### Do there exist pseudo-reductive (but not reductive) groups of small dimension?

I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is ...

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**3**answers

957 views

### Small index subgroups of SL(3,Z)

I would like to know the smallest index subgroups of SL(3,Z).
The smallest I could find has even entries $a_{3,1}$ and $a_{3,2}$,
along the bottom row. I could not figure out whether there are
...

**1**

vote

**1**answer

83 views

### on z-extensions

Let $G$ a group split over a local field $F$.
We call a $z$-extension a group $G'$ such that $G'_{der}$ is simply connected, $G'$ is a central extension of $G$ by a central torus $Z$.
Can we find a ...

**0**

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**0**answers

162 views

### minuscule representations and classical groups

Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...

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votes

**3**answers

381 views

### Dimension of Unipotent Radicals

A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...

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**1**answer

560 views

### General Bruhat decomposition (with parabolic not necessarily Borel)

Here is the general Bruhat decomposition (which I have seen in various paper but never with a proof or a complete reference).
Let $G$ be a split reductive group, $T$ a split maximal torus and $B$ a ...

**1**

vote

**1**answer

321 views

### center of the centralizer of semisimple element

Let $G$ be an adjoint group over an algebraically closed field $k$ and $s\in G$ a semisimple element.
Let $H=C_{G}(s)^{0}$ the neutral component of the centralizer of $s$. Do we have that the center ...

**1**

vote

**1**answer

270 views

### Questions about multiplicative homomorphism of $\mathbb{R}$

Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.
Question 1. Whether that $f$ is surjective implies that $f$ is injective?
...

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**2**answers

373 views

### Center of the algebraic group G_{\mathbb{R}} for a centerless G

This must be an easy question but I don't have a good argument for it and have not found a counterexample: Let $G$ be a connected semisimple algebraic group over $\mathbb{Q}$ such that the center of ...

**0**

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**1**answer

141 views

### subgroups of a $p$-solvable group and complete reducibility

1.
Let $G$ be a $p$-solvable group and $V$ be a finite dimensional
faithful $kG$-module, where the characteristic of $k$ is $p$. But
$V$ is not a semisimple $kG$-module. For every $n\geq 0$, we ...

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votes

**0**answers

95 views

### on a decomposition lemma in adelic groups

Let X a curve over an algebraically closed k.
Fix $x$ and $y$ two distinct closed points of X.
Let G be a connected reductive group over k.
We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal ...

**0**

votes

**1**answer

200 views

### For which semisimple element $s$ in finite group of lie type centralizer $C_{G}(s)$ of $s$ is a Levi subgroup ?

Let $G$ be a finite simple group of lie type. Let $s$ be a semisimple element lying in maximal torus $T_{w}$ for $w\in W$ where $W$ is the Weyl group of $G$. Can we say that $C_{G}(s)$ is Levi ...

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votes

**1**answer

581 views

### Groups becoming algebraic groups

Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that:
(1) the set of $k$-points has the structure of a group.
(2) for any $g\in G$ the ...

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votes

**1**answer

375 views

### Confusing Point in Proof: Semisimple Automorphism Fixes Torus

I am reading a proof on p.51 of Robert Steinberg in his book "Endomorphisms of Algebraic Groups" and I am having a bit of difficulty understanding one point in the proof.
The setting is as follows. ...

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votes

**1**answer

300 views

### On the Steinberg section

Let $\chi:G\rightarrow T/W$ the Steinberg map. I assume that G is simply connected. Then $T/W=\mathbb{A}^{r}$ and Steinberg constructed a section to this map given by
...

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vote

**1**answer

174 views

### Describing a matrix group (with integer coefficients) through conditions on the coefficients.

I'm wondering if there's always a (not too complicated?) way to characterize a matrix group by conditions on the coefficients.
I know if I'm dealing with matrix groups over a field, then it's sort of ...

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**2**answers

421 views

### Regular elements in the torus of a group of Lie type

Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...

**5**

votes

**5**answers

558 views

### Commutator of algebraic subgroups is connected

Let $G$ be an algebraic group over an
algebraically closed field. If $H$ and
$K$ are closed subgroups and one of
them is connected, then their
commutator $[H,K]$ is also connected.
Is ...

**0**

votes

**1**answer

284 views

### For an algebraic group acting on a variety, why are orbits representable?

I suspect this is really obvious, but I'm not seeing it.
For an algebraic group $G$ acting on a variety $V$, and for a point $x \in \text{hom}(\text{Spec}(K),V)$, we define the orbit $G(x)$ to be ...