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I have two questions. Thanks for any comments.

Suppose $S$ is a simple group of Lie type in characteristic p. Also suppose that $G=Aut(S)$.

1) Is there any reference for conjugacy class of element $g$ of $G$? (GLS has some results for the case p=2) In particular when $g$ is a field automorphism, is there any result?

2) Is there a field automorphism of order p, when p is the characteristic of $S$?

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    $\begingroup$ As to 2, there are certainly field automorphisms of order $p$ for some values of $q.$ For example when $q = p^{p},$ the automorphism $a \to a^{p}$ of ${\rm GF}(q)$ has order $p.$ $\endgroup$
    – Geoff Robinson
    Nov 30, 2014 at 18:51
  • $\begingroup$ There are loads of references for this sort of thing - for $PSL_n(q)$ you can use rational forms, for the other classical groups there are variations on rational forms due to MacDonald and also to Wall. For groups of Lie type in full generality I would go to Carter's Finite groups of Lie type. The ATLAS is also very helpful for individual small groups. $\endgroup$
    – Nick Gill
    Dec 1, 2014 at 22:35
  • $\begingroup$ For field automorphisms: the definition of a field automorphism of a Chevalley group is that it is an $Aut(S)$ conjugate of en element of $\Phi_S$ where $\Phi_S$ is the cyclic subgroup of "entry-by-entry automorphisms". So the conjugacy structure is a triviality. For Steinberg groups and the others the definitions are similar (see Def. 2.5.13 of GLS3). $\endgroup$
    – Nick Gill
    Dec 1, 2014 at 22:39

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