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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122 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 ...
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0answers
47 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 ...
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1answer
65 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
150 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
168 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?
31
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2answers
474 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
84 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
56 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 ...