# Questions tagged [algebraic-groups]

For questions about groups which have additional structure as algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. These include both linear algebraic groups as well as abelian varieties. Consider using with the (group-theory) tag.

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### Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$

Let $G$ be a simply connected absolutely simple group of one of the types $^1{\sf A}_{n-1}$ (inner) or $^2{\sf A}_{n-1}$ (outer) over a field $k$. All such groups are described on page 55 of Tits, ...
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### maximal subgroup of the general linear group

Maybe I'm being silly. As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, is the conformal orthogonal group $CO_{2n}$ a maximal subgroup of $GL_{2n}$(or $CO_{2n+1}$ ...
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### $\mathbb{G_a}$ is being linear algebraic group, but first diagram does not commute?

Question: Why they are all assume it trivial, but I cannot make it commute the diagram in the first axiom of being, linear algebraic group (for $A=k[s]$). Definition 3.48.(ref.Mukai An Introduction to ...
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### Order of Chevalley group

Let $G$ be a semisimple algebraic group defined over $\mathbb{Q}$ (simply connected if it helps). Then for almost all (maybe all) primes $p$, we can make sense of the group $G(p)$ (formed by reducing ...
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### Is $\operatorname {GL}(n)\times \operatorname{Gr}(m,n)\to \operatorname{Gr}(m,n)$ closed?

Suppose $k$ is a field, and $m<n$ are nonnegative integers. Let $\operatorname{Gr}(m,n)$ be the Grassmannian (whose points are $m$-dimensional subspaces of a $n$-dimensional linear space). Then we ...
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### Show that the identity component $G^\circ\sqsubseteq G$ (i.e., is characteristic) in a linear algebraic group $G$.

This is Exercise 7.6.6 of Humphreys', "Linear Algebraic Groups". The Question: Show that the identity component $G^\circ$ of a linear algebraic group $G$ is a characteristic subgroup. The ...
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### Is a closed, nonempty subset of a (linear) algebraic group $G$ that is closed under taking products a subgroup of $G$? [duplicate]

This is a question based on Exercise 7.6.5 of Humphreys', "Linear Algebraic Groups". For a solution to that particular problem, see: A closed subset of an algebraic group which contains $e$ ...
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### Is the quotient group of $2$-th root of unity compact? [closed]

Let $F$ be a number field and $\mathbb{A}$ its adele ring. Let $\mu_2$ be the group scheme of 2-th root of unity defined over $F$. I guess that $\mu_2(F) \backslash \mu_2(\mathbb{A})$ is compact ...
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1 vote
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### A Question about Matrix Computation in GL4

Let us consider $G=GL_4(k)$, where $k=\overline{\mathbb{F}_p}$. Consider the set $S$ of the following kind of elements in $G$:  \left( \begin{matrix} 1 & 0 & \ast &\ast \\ 0 & 1 &...
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### Does an arbitrary algebraic group G possess a unique largest normal solvable subgroup?

On page 125 of Humphreys' "Linear Algebraic Groups," it states the following: "An arbitrary algebraic group G possesses a unique largest normal solvable subgroup." However, I am ...
### If a hom. $\phi:G\to H$ of diagonalisable linear algebraic groups is injective, then the induced hom. $\phi^*:X^*(H)\to X^*(G)$ is surjective
This is Exercise 3.2.10(2) of Springer's book, "Linear Algebraic Groups (Second Edition)". According to Approach0, it is new to MSE. The Question: Let $\phi:G\to H$ be a homomorphism of ...