Classification of subgroups of symplectic groups over finite fields containing a transvection

• Autorzy: Arias-de-Reyna, S. , Dieulefait, L. , Wiese, G.
• Języki publikacji: EN

Abstrakty

EN

In this note, we give a self-contained proof of the following classification (up to conjugation) of finite subgroups of GSpn(Fl) 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(Fl). This result is for instance useful for proving ‘big image’ results for symplectic Galois representations.

Słowa kluczowe

PL

teoria grup; klasyfikacja podgrupy grup liniowych

EN

sympletic group over a finite field; transvection; classification of subgroups of linear groups

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2016
• Tom: Vol. 49, nr 2
• Strony: 129--148
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Arias-de-Reyna, S.

• Université Du Luxembourg Faculté Des Sciences, De La Technologie Et De La Communication 6, Rue Richard Coudenhove-Kalergi L-1359 Luxembourg, Luxembourg,

autor

Dieulefait, L.

• Departament D’àlgebra I Geometria Facultat De Matemàtiques Universitat De Barcelona Gran Via De Les Corts Catalanes, 585 08007 Barcelona, Spain,

autor

Wiese, G.

• Université Du Luxembourg Faculté Des Sciences, De La Technologie Et De La Communication 6, Rue Richard Coudenhove-Kalergi L-1359 Luxembourg, Luxembourg,

Bibliografia

• [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 PSp(2n; F), 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.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.