# 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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### objects with non-alternating simple symmetry group

The usual soccer ball has as its group of symmetries (counting rotations only) the alternating group $A_5$. Is there an example of a well known object whose group of symmetries is a finite simple ...
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### centralizer in Magma

Sorry that I am new to Magma. I have a question: I can construct the group of Lie type $3E_6(C)$ and an element $a$ in it. However I cannot have Magma compute the centralizer of $a$ as it gives ...
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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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### 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?
1 vote
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### 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 ...
1 vote
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### 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 ...
1 vote
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### $Spin_n(q)$, $SO_n(q)$, $\Omega_n(q)$ 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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### $\mathbb{F}_q$-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 ...
1 vote
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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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### 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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### 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: ... 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 ...