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As a follow-up to my previous question, I need to investigate the action of certain outer automorphisms of $D_4(q)$ on specific $2$-subgroups (the defect groups of its blocks, if anyone is interested). This is completely outside my comfort zone, and of course I want to learn more about it, but I am a bit lost. I am not necessarily looking for an answer, a reference to start my own investigation would also suffice. First question is:
1) I see mentioned everywhere (CFSG for example) that $D_n(q)$ is a simple group and it corresponds to a classical group. But... which one? The Magma Handbook says it is $P \Omega^+(2n,q)$, but what about $P \Omega^-(2n,q)$? Isn't it also a simple group? And if yes, where does it fit in the classification? What is a bit unclear to me is precisely which classical groups correspond to the four types of Dynkin diagram $A, B, C, D$.

Second question is:
2) Where can I find a precise characterization of these automorphisms, to be able to see how they act on particular elements? I am guessing most literature in the language of classical groups. I looked at the very useful "The Finite Simple Groups" by Wilson but I am looking for a more precise definition of the outer automorphisms.

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The correspondence between the Chevalley groups of Lie type and the finite classical simple groups is as follows:

$A_n(q)$: ${\rm PSL}(n+1,q)$

$^2A_n(q)$: ${\rm PSU}(n+1,q)$

$B_n(q)$: ${\rm P \Omega}(2n+1,q)$

$C_n(q)$: ${\rm PSp}(n,q)$

$D_n(q)$: ${\rm P \Omega}^+(2n,q)$

$^2D_n(q)$: ${\rm P \Omega}^-(2n,q)$.

But surely that information is in the ATLAS? The untwisted groups of Lie type over a finite field of order $q$ arise as the fixed points of a Frobenius (field) automorphism $x \mapsto x^q$ acting on the Lie group over an algebraically closed field.

The twisted groups like $^2A_n(q)$ arise as the fixed points of the product of this Frobenius automorphism and a graph automorphism. Note that the natural representation of ${\rm PSU}(n,q)$ is as a ssubgroup of ${\rm PSL}(n,q^2)$, and ${\rm P \Omega}^-(2n,q)$ is a also a subgroup of ${\rm P \Omega}^+(2n,q^2)$.

The book by Kleidman and Liebeck on the maximal subgroups of the finite classical groups has a very detailed description of their automorphisms in one of the earlier chapters, and I have heard that a lot of people use it as a reference for that.

I believe also that one of the volumnes in the sequence of books by Gorenstein, Lyons, and Solomon on the classification of finite simple groups is devoted to properties of the simple groups, including their automorphism groups. It might be Volume 3.

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  • $\begingroup$ Thank you! Shouldn't it be $A_n(q) : PSL(n+1,q)$? $\endgroup$ – AnalysisStudent0414 May 9 at 10:30
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    $\begingroup$ Yes - and similarly for ${\rm PSU}$. $\endgroup$ – Derek Holt May 9 at 10:59

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