Journal of Lie Theory 26 (2016), No. 1, 193--218
Copyright Heldermann Verlag 2016
θ-Semisimple Classes of Type D in PSLn(q)
Giovanna Carnovale
Dip. di Matematica, Torre Archimede - via Trieste 63, 35121 Padova, Italy
carnoval@math.unipd.it
Agustín García Iglesias
FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
aigarcia@famaf.unc.edu.ar
[Abstract-pdf]
\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.
Keywords: Hopf algebras, twisted conjugacy classes, finite simple groups.
MSC: 16W30
[ Fulltext-pdf (460 KB)] for subscribers only.