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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5
votes
2answers
200 views

How to calculate $|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$?

the answer should be $$|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|=N^{3}\prod_{p|N}(1-{1 /p^2})$$ But first how to prove $$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$
15
votes
2answers
292 views

Connectedness of centralizer exercise

I'm having trouble understanding connectedness from a group theoretic perspective. Let $G$ be the symplectic group of dimension 4 over a field $K$, $$G = \operatorname{Sp}_4(K) = \left\{ A \in ...
4
votes
2answers
139 views

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 ...
4
votes
2answers
247 views

Reference request for algebraic Peter-Weyl theorem?

It seems that, for $GL_n$, and possibly for something like complex reductive groups $G$ in general, there's an algebraic version of the Peter-Weyl theorem, which might say that the coordinate ring of ...
2
votes
1answer
54 views

Factor/Quotient Group $M/\mathbb{Z}$

I've posted another question here a few days ago asking what a factor/quotient group is because I couldn't wrap my mind around it. Although I have an idea what it means, I still don't fully understand ...
3
votes
1answer
218 views

If a subgroup of an algebraic group is solvable, is its closure necessarily solvable?

$G$ is an algebraic group, and $H$ is a subgroup which is solvable. $\overline{H}$ is its closure in $G$. Then $\overline{H}$ is also a subgroup of $G$. Is it also solvable? For any algebraic group ...
5
votes
0answers
104 views

Is there a classification of finite abelian group schemes?

There is a well-known classification of finite abelian groups into products of cyclic groups. What about finite abelian group schemes, where we may put in the qualifiers "affine", "etale", or ...
6
votes
2answers
164 views

For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = \overline{[H,H]}$? What about algebraic groups?

When discussing with awllower about this question, I begin to think about another one: For a topological group $G$ and a subgroup $H$, is it true that $[\overline{H}, \overline{H}] = ...
3
votes
1answer
71 views

Geometric difference between two actions of $GL_n(\mathbb{C})$ on $G\times \mathfrak{g}^*$

Let $G=GL_n(\mathbb{C})$. Scenerio 1: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ g.(x,y)=(gx,y). $$ Scenerio 2: Let $G$ act on $T^*(G)=G\times \mathfrak{g}^*$ by $$ ...
3
votes
0answers
229 views

question regarding Waterhouse, affine group schemes

Excerpt from Waterhouse, 14.4 Structure of Finite Connected groups. Thm. Let $A$ represent a finite connected group scheme over a perfect field of characteristic $p$. Then $A$ has the form $k[X_1, ...
1
vote
1answer
33 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 ...
1
vote
2answers
149 views

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 ...