Questions tagged [groups-of-lie-type]

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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Generation of $SU(n,q^2)$ by subgroups isomorphic to $SU(2,q^2)$

Throughout, let $q$ be an odd prime power. Let $GF(q^2)$ be the field with $q^2$ elements. My question concerns the generation of the special unitary group $SU(n,q^2)$, where $n \ge 2$, by certain ...
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Direct products inside finite simple groups

For a finite group $G$, let $d(G)$ be the largest $k$ such that $G$ admits a subgroup isomorphic to a direct product of $k$ non-trivial groups. I am interested in families $G_n$ of finite simple ...
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1answer
49 views

What is the order and structure of Aut(SL$_2(p)$)?

Here $p$ is a prime. We know that for $Z(SL_2(p)) = \{ \pm I \}$ and $\lvert SL_2(p)\rvert= p^3-p$ so there are $\frac{p^3-p}{2}$ inner automorphisms. What is the outer automorphism group?
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1answer
51 views

Finite simple group has order a multiple of 3?

Checking the list of finite simple groups, it seemed to me that all groups have order a multiple of $3$. This clear for alternating groups and checked case by case for sporadic groups. For groups of ...
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30 views

What is meant here by “element of the prime field corresponding to $a \in \mathbb{Z}$”?

I'm reading about the construction of Chevalley groups over an arbitrary field K. Carter originally introduces an automorphism on the Chevalley basis $\{h_r,r \in \Pi; \quad e_r, r \in \Phi \}$ and ...
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43 views

$Spin_n(q)$, $SO_n(q)$, $\Omega_n(q)$ and their projective images

I am studying about the structure of orthogonal groups and struggling to understand the relations between groups in the title: From algebraic groups point of view, it is known that $Spin_n(q)$ is ...
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43 views

$\mathbb{F}_q$-rational elements in unipotent classes of simple algebraic group in positive characteristic

Sorry in advance if this question is trivial or trivially false. I haven't managed to find a satisfactory proof (or reference of one), or a counterexample for it. Let $k$ be the algebraic closure of ...
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Reference for Finite groups of Lie type

I want to learn about finite groups of Lie type, but I have no experience in Lie groups. What could be a good reference-book/notes for this?
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244 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 \vec{\sigma}\vec{n}/...
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1answer
166 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 ...
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2answers
740 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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88 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
236 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 $K$ ...
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1answer
757 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 ...
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1answer
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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?
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741 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: ...
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1answer
199 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 ...
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133 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 ...