# Tagged Questions

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### Weil restriction for schemes

I'm trying to understand the Weil restriction of a scheme (since I'm reading a paper which uses it). I'm even having troubles trying understanding the following "toy" example. Toy example. Let $X$ be ...
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### On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
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### The set of one-parameter subgroup of the Multiplicative group $G_m$ is Z

Let $G_m= k^{*}=k-{0}$ be the multiplicative group. We know this is an Algebraic group also. How does one prove any algebraic group morphism $G_m \rightarrow G_m$ is of the form $t \mapsto t^{n}$ for ...
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### Felix Klein's view on algebraic geometry

I think, as a first approach one would say that a geometry on a set $X$ is given by an inner product on $X$. Klein then links geometry to group theory by identifying a geometry on $X$ with a group of ...
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### Find a closed subset of an algebraic group, closed under products, which does not contain $e$.

The accepted answer for this question proves the following statement: If $S$ is a closed subset of an algebraic group $G$ which contains $e$ and is closed under taking products in $G$, then $S$ is ...
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### definition of semidirect product on two projective spaces

What is the definition of semidirect product on two projective spaces $\mathbb CP^1 \rtimes\mathbb CP^2$ I can not undrestand it.
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### Coset variety of an algebraic group

Let $k$ be an algebraically closed field of characteristic $p\geq0$. An affine algebraic group $G$ is an affine algebraic variety (a Zariski closed subset of $k^m$ for some $m$) such that ...
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### How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
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### parabolic subalgebra

Let $G$ be a semisimple lie group, a parabolic subgroup of $P$ is a connected subgroup that contains a conjugate of $B$, (which $B$ is Borel subgroup of $G$) then I can not see why lie algebra of $P$ ...
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### Exceptional isomorphisms of classical algebraic groups

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. An affine algebraic group $G$ is an affine variety over $k$ with a group structure such that multiplication and inversion are ...
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### relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
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### Picard group of a Affine scheme

How do we define a Picard group of an Affine scheme? Is there way to define as for commutative ring? Thanks
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### Are simple algebraic groups absolutely simple?

Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ ...
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### Definitions in a Theorem of Lang

I'm trying to understand the following theorem due to Serge Lang (Algebraic Groups over Finite Fields, 1956, Theorem 2): Let $p$ be a prime, and let $k$ be a finite field of $q=p^n$ elements. Let ...
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### how to make factorization by a group action

any algebra and numerical example for Projectivization http://en.wikipedia.org/wiki/Projectivization which book or paper teaching this
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### Automorphism group of the elliptic curve $y^2 + y = x^3$

Consider the elliptic curve $E : y^2+y = x^3$ over $\overline{\mathbb{F}_2}$. It has the biggest automorphism group $G$ among all elliptic curves, namely with order $24$. What is the structure of $G$? ...
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### Factor group of quaternion group

I have $Q_8=\{I,A,A^2,A^3,B,AB,A^2B,A^3B\}$ where $A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$ Now I take a look at $Q_8/N$ ...
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### Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
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### For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
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### Finite automorphism groups of $\mathbb{P}^1$

I would like to know all finite groups of $\operatorname{Aut}(\mathbb{P}^1)$. I am aware of that any automorphism of $\mathbb{P}^1$ is given by Möbius transformation $$z\mapsto\frac{az+b}{cz+d}$$ ...
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### groups acting on curves

Can anybody help me to prove the following statement ? *Let $X$ be a smooth, connected projective curve defined over a number field $k \subset \mathbb{C}$. Let $G$ be a finite group acting on $X$. ...
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### Any affine algebraic group is linear.

It is a well-known result that any affine algebraic group is a closed subgroup of some $\mathrm{Gl}_n(\Bbbk)$. However, I would like to see a proof for that, so I looked it up in various books, more ...
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### Show that affine transformations are associative

I'm having trouble with the associativity part of showing that $\text{Aff}(\mathbb{R}^2)$ is a group. Recall that if $\left[ A, \overline{r} \right]$ denotes an affine transformation (where $A$ is ...
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### Orbit-stabilizer theorem for Lie groups?

Let $G$ be a finite-dimensional lie group, with a transitive action on the points of a smooth finite-dimensional manifold $S$. Let $p$ be some point of $S$ and let $T$ be the stabilizer of $p$ in $G$. ...
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### Fencing the Group size,and its implication to Finiteness of Tate-Shafarevich Group

This question is an interesting one,not like my previous one. Can we judge the size of a Quotient Group by seeing the size of its constituents ? To add something ,Suppose consider a group ...
### Is it true that $\forall G\leq Aut(\mathbb{P}^1)=PGL_2(\mathbb{C})$ the map $\mathbb{P}^1\rightarrow \mathbb{P}^1/G$ is defined over $\mathbb{Q}$?
Is it true that for every finite $G\leq Aut(\mathbb{P}^1_{\mathbb{C}})=PGL_2(\mathbb{C})$ the morphism $\mathbb{P}^1_{\mathbb{C}}\rightarrow \mathbb{P}^1_{\mathbb{C}}/G$ descends as a morphism (not ...
Let $k$ be an algebraically closed field and let $G\leq\rm{GL}_n(k)$. Assume that $M<G$ is a maximal subgroup (in the abstract group sense). Denote by $\bar{G}^Z$ the Zariski closure of $G$ in ...