A group closely related to the group G(k) of rational points of a reductive linear algebraic group G with values in the field k. Not to be confused with (lie-groups).

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algebra operations

If $[x]=x^{i}\sigma^{i}$, Find an alternative form for the product $$ x^{i}n^{j}\sigma^{k}\epsilon^{ijk}=x^{i}(\vec{n} \times \vec{\sigma})^{i} $$ that has to be more compact than $$ ...
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0answers
51 views

Lie Algebra SU(2)

Given a two dimensional Hilbert-space, $\mathcal{H}$, and a vector $\eta \in \mathcal{H}$, of this space, if $\eta$ transforms in SU(2) like this, $$\eta \rightarrow e^{(-i\alpha ...
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1answer
31 views

What exactly is the group Omega(n,q) in MAGMA?

Let $n>2$ and $q$ be a prime power. In MAGMA I'm having a lot of trouble identifying the group Omega(n,q). I'm trying to use a source that asserts that it is the group of $n\times n$ orthogonal ...
2
votes
2answers
235 views

Sylow $p$-subgroups of finite simple groups of Lie type

I need some information about the Sylow $p$-subgroups, and their normalizers, (specially their sizes), of a finite simple group of Lie type over a finite field (not necessarily algebraic closed) of ...
1
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0answers
60 views

concrete examples of finite groups of lie type

I was told that there were types of finite groups of lie types, such as $A_l,l\geq 1$, $^2A_l, l \geq 2$, $B_l, l \geq 2$, $^2B_2$ and so on. My problem is that if there are any concrete examples of ...
1
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1answer
117 views

non-split extension of the simple group $L_3(4)$

I would like to know the structure of the groups $L_3(4).C_2$ and $L_3(4).C_{11}$. (By $C_n$ I mean the cyclic group of order $n$ and by $G=K.L$ I mean the non-spli extension of $K$ by $L$, were $K$ ...
11
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1answer
186 views

Abelian subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number, and let $k=\mathbb{F}_p$ be the field of $p$ elements. Let $G=GL_n(k)$. We know that $$|G|=\prod_{i=0}^{n-1}(p^n-p^i)=p^{\binom{n}{2}}\prod_{i=0}^{n-1}(p^{n-i}-1)$$ so ...
6
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1answer
243 views

Why is $PGL_2(5)\cong S_5$?

Why is $PGL_2(5)\cong S_5$? And is there a set of 5 elements on which $PGL_2(5)$ acts?
32
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2answers
523 views

Are there/Why aren't there any simple groups with orders like this?

The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this: ...
3
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1answer
94 views

List of finite groups of Lie type and their BN-pairs

as the title states I am looking for a list of classical groups (or perhaps finite groups of Lie type) and their respective BN-pairs (or isomorphism type of the respective Weyl group). A quick Google ...
2
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0answers
69 views

How does the Frobenius map permute the roots

How can a Frobenius map permute the roots of an algebraic group? According to Carter (in Finite groups of Lie type), a root subgroup $X_{\alpha}$ is the 1-dimensional unipotenet subgroup giving rise ...