Classification of Subgroups of Symplectic Groups Over Finite Fields Containing a Transvection

S. Arias-de-Reyna 1 , L. Dieulefait 2  and G. Wiese 1
  • 1 UNIVERSITÉ DU LUXEMBOURG FACULTÉ DES SCIENCES, DE LA TECHNOLOGIE ET DE LA COMMUNICATION 6, RUE RICHARD COUDENHOVE-KALERGI L-1359 LUXEMBOURG, LUXEMBOURG
  • 2 DEPARTAMENT D’ÀLGEBRA I GEOMETRIA FACULTAT DE MATEMÀTIQUES UNIVERSITAT DE BARCELONA GRAN VIA DE LES CORTS CATALANES, 585 08007 BARCELONA, SPAIN

Abstract

In this note, we give a self-contained proof of the following classification (up to conjugation) of finite subgroups of GSpn(F̅) containing a nontrivial transvection for ℓ ≥ 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Spn(F̅). This result is for instance useful for proving ‘big image’ results for symplectic Galois representations.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] S. Arias-de-Reyna, L. Dieulefait, S. W. Shin, G. Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties, Math. Ann. 361(3) (2015), 909-925.

  • [2] S. Arias-de-Reyna, L. Dieulefait, G. Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations, Trans. Amer. Math. Soc., in press, 2016.

  • [3] S. Arias-de-Reyna, L. Dieulefait, G. Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image, Pacific J. Math. 281(1) (2016), 1-16.

  • [4] E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York-London, 1957.

  • [5] L. E. Dickson, Linear Groups: With an Exposition of the Galois Field Theory, with an introduction by W. Magnus, Dover Publications Inc., New York, 1958.

  • [6] W. M. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc. 248 (1979), 347-379.

  • [7] S. Z. Li, J. G. Zha, On certain classes of maximal subgroups in PSpp2n; Fq, Sci. Sinica Ser. A 25 (1982), 1250-1257.

  • [8] H. H. Mitchell, Determination of the ordinary and modular ternary linear groups, Trans. Amer. Math. Soc. 12 (1911), 207-242.

  • [9] H. H. Mitchell, The subgroups of the quaternary Abelian linear group, Trans. Amer. Math. Soc. 15 (1914), 379-396.

  • [10] A. Wagner, Groups generated by elations, Abh. Math. Sem. Univ. Hamburg 41 (1974), 190-205.

OPEN ACCESS

Journal + Issues

Demonstratio Mathematica, founded in 1969, is a fully peer-reviewed, open access journal that publishes original and significant research works and review articles devoted to functional analysis, approximation theory, and related topics. The journal provides the readers with free, instant, and permanent access to all content worldwide (all 53 volumes are available online!)

Search