θ-Semisimple Classes of Type D in PSL_n(q)

G. Carnovale ; A. García Iglesias

Journal of Lie theory, Tome 26 (2016) no. 1, pp. 193-218

Résumé

\def\F{{\Bbb F}} \def\N{{\Bbb N}} Let $p$ be an odd prime, $m\in \N$ and set $q=p^m$, $G={\rm PSL}_n(q)$. Let $\theta$ be a standard graph automorphism of $G$, $d$ be a diagonal automorphism and ${\rm Fr}_q$ be the Frobenius endomorphism of ${\rm PSL}_n(\overline{\F_q})$. We show that every $(d\circ \theta)$-conjugacy class of a $(d\circ \theta,p)$-regular element in $G$ is represented in some ${\rm Fr}_q$-stable maximal torus of ${\rm PSL}_n(\overline{\F_q})$ and that most of them are of type D. We write out the possible exceptions and show that, in particular, if $n\geq 5$ is either odd or a multiple of $4$ and $q>7$, then all such classes are of type D. We develop general arguments to deal with twisted classes in finite groups.

Détail

Classification : 16W30
Mots-clés : Hopf algebras, twisted conjugacy classes, finite simple groups