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Is there a convenient list of the maximal abelian subgroups of the projective semilinear group $\mathrm{P\Gamma L}_3(K) \cong \mathrm{PGL}_3(K) \rtimes \mathrm{Gal}(K)$ for $K$ a finite field?

This is related to this previous question about maximal abelian subgroups of $\mathrm{GL}_n$: Maximal abelian subgroup of general linear groups. My question is easier in that $n = 3$, but harder in that I am mod $K^\times$ and $\rtimes \mathrm{Gal}$, which is irritating enough that it would be good to have a reference if there is one.

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  • $\begingroup$ I don't know of an easy reference, sorry. Lots of the elements in $P\Gamma L_3(K)$ have abelian centralizers I think. This is useful because (a) such centralizers must be maximum abelian subgroups; (b) if $g$ is an element with an abelian centralizer, then the only maximum abelian subgroup containing $g$ will be $C_G(g)$. You probably know this but anyway... $\endgroup$
    – Nick Gill
    Commented Dec 4, 2020 at 14:06
  • $\begingroup$ Thanks for your comment, it is helpful. Incidentally I have learned that there is a "Dembowski--Piper classification" of large abelian subgroups of collineation groups of projective planes (desarguesian or not). $\endgroup$
    – Sean Eberhard
    Commented Dec 5, 2020 at 10:36
  • $\begingroup$ Ah, well that sounds just the ticket! $\endgroup$
    – Nick Gill
    Commented Dec 7, 2020 at 10:32

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