# Tagged Questions

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

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### A closed subset of an algebraic group which contains $e$ and is closed under taking products is a subgroup of $G$

A closed subset of an algebraic group which contains $e$ and is closed under taking products is a subgroup of $G$. Denote this set as $X$. If the condition of $X$ being closed is dropped, this ...
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### Diagonalizable linear algebraic group is isomorphic to $(\mathbb{C}^*)^r\times A$, for some finite abelian group $A$

I have three questions about algebraic groups. Let $D$ be a linear algebraic group. Then the following are equivalent: $D$ is diagonalizable. $\mathop{Hom}(D,\mathbb{C}^*)$ is finitely generated ...
Let $$s = \begin{pmatrix} \lambda_1 & & & 0\\ & \lambda_2 & \\ & & \ddots \\ 0& & & \lambda_n \end{pmatrix}$$ be a diagonal invertible matrix. Let $G = \textrm{... 1answer 52 views ### Non-separated quotient of separated scheme I am reading Mumford's GIT book. I found the following claim there. Let$X$be an algebraic variety. Let$G$be an algebraic group acting on$X$. Then the categorical quotient of$X$by$G$may be ... 1answer 47 views ### For an element$x$in an algebraic group$G$, why do we have$\mathscr{L}(C_G(x))\subset\mathfrak{c}_{\mathfrak{g}}(x)$? I'm reading Humphreys' Linear Algebraic Groups, trying to understand the following argument found on the top of pg. 76. Let$G$be an algebraic group over some field$k$, with$x\in G$. Let$\...
Suppose some maximal torus $T$ of $G$ is $C_G(T)$, then the set of semisimple elements for which $C_G(s)$ is a torus contains a nonempty open set. Such element are called regular semisimple. I want ...