Abstract
We prove that if u is a unipotent element of a connected reductive algebraic group G over\(\bar F_2 \), there exists an involution σ in G such that σuσ=u−1. We use this result to determine the group of lattice automorphisms of G, when G is simple.
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References
Aschbacher, M., Seitz, G.M.: Involutions in Chevalley groups over fields of even order, Nagoya Math. J.63, 1–91 (1976), Nagoya Math. J.72, 135–136 (1978)
Bala, P., Carter, R.W.: Classes of unipotent elements in simple algebraic groups I, Math. Proc. Cambridge Philos. Soc.79, 401–425 (1976)
Bala, P., Carter, R.W.: Classes of unipotent elements in simple algebraic groups II, Math. Proc. Cambridge Philos. Soc.80, 1–18 (1976)
Carter, R.W., Elkington, G.B.: A note on the parametrization of conjugacy classes, J. Algebra20, 350–354 (1972)
Carter, R.W.: Simple groups of Lie type, John Wiley, London (1972)
Carter, R.W.: Finite groups of Lie type, John Wiley, London (1985)
Costantini, M.: Automorphisms and autoprojectivities of certain algebraic groups, Rend. Acc. Naz. delle Scienze detta dei XL, 110, Vol. XVI, fasc. 7, 99–114 (1992)
Costantini, M.: On the lattice automorphisms of certain algebraic groups, Rend. Sem. Mat. Univ. Padova, Vol.90, 141–157 (1993)
Costantini, M.: The group of autoprojectivities of\(SL_3 \left( {\bar F_p } \right)\) and\(PGL_3 \left( {\bar F_p } \right)\), J. Algebra171, 113–151 (1995)
Dieudonné, J.: Le géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete,5, Springer, Heidelberg (1955)
Elkington, G.B.: Centralizers of unipotent elements in semisimple algebraic groups, J. Algebra23, 137–163 (1972)
Gow, R.: Products of two involutions in classical groups of characteristic 2, J. Algebra71, 583–591 (1981)
Metelli, C.: Sugli isomorfismi reticolari di PSL2(pf), Rend. Acc. Naz. Lincei Cl. Scienze, s. VIII, vol. XLVII, fasc.6 (1969)
Mizuno, K.: The conjugate classes of Chevalley groups of type E6, J. Fac. Sci. Univ. Tokyo,24, 525–563 (1977)
Mizuno, K.: The conjugate classes of unipotent elements of the Chevalley groups of type E7 and E8, Tokyo J. Math.,3, 391–461 (1980)
Shinoda, K.: The conjugacy classes of Chevalley groups of type (F4) over finite fields of characteristic 2, J. Fac. Sc. Univ. Tokyo,21, 133–159 (1974)
Spaltenstein, J.N.: Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics,946, Springer, New York (1982)
Steinberg, R.: Conjugacy classes in algebraic groups, Lecture Notes in Mathematics,366, Springer, Heidelberg (1974)
Stuhler, U.: Unipotente und nilpotente Klassen in einfachen Gruppen und Lie algebren vom Type G2, Indag. Math.33, 365–378 (1971)
Völklein, H.: On the lattice automorphisms of the finite Chevalley groups, Indag. Math.89, 213–228 (1986)
Völklein, H.: On the lattice automorphisms of SLn(q) and PSLn(q), Rend. Sem. Mat. Univ. Padova76, 207–217 (1986)
Zacher, G.: Sul reticolo dei sottogruppi del gruppo simmetrico, Rend. Ist. di Matem. Univ. di Trieste Vol.IX fasc. 1–2, 122–126 (1977)
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Costantini, M. The lattice automorphisms of simple algebraic groups over\(\bar F_2 \) . Manuscripta Math 91, 1–16 (1996). https://doi.org/10.1007/BF02567936
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DOI: https://doi.org/10.1007/BF02567936