Abstract
Let b be a nondegenerate symplectic form on a vector space V over a finite field. It is well-known that every intermediate group between Sp(V,b) and ΓSp(V,b) (i.e. the isometry and the semisimilarity groups of b, respectively) is Rank 3 in its action on the projective space P(V). We prove that this property characterizes such subgroups of ΓSp(V,b) when the dimension of V is greater than 2.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. S. Abhyankar Symplectic groups and permutation polynomials, Part I
S. S. Abhyankar and N. F. J. Inglis, Thoughts on Symplectic groups and Symplectic equations
E. Artin, The orders of linear groups, Communications on Pure and Applied Mathematics, vol, 8 (1955), pages 355–365.
G. D. Birkhoff and H. S. Vandiver, On the integral divisors of a n − b n, Annals of Mathematics, vol. 5 (1904), pages 173–180.
P. J. Cameron and W. M. Kantor, 2-Transitive and antiflag transitive collineation groups of finite projective spaces, Journal of Algebra, vol. 60 (1979), pages 384–422.
L. E. Dickson, On the cyclotomic function, American Mathematical Monthly, vol. 12 (1905), pages 86–89.
W. Feit, On large Zsigmondy primes, Proceedings of the American Mathematical Society, vol. 102 (1988), pages 29–36.
D. G. Higman and J. E. McLaughlin, Rank 3 subgroups of symplectic and unitary groups, Crelle Journal, vol. 218 (1965), pages 174–189.
D. Perin, On collineation groups of finite projective spaces, Math. Zeit., vol. 126 (1972), pages 135–142.
D. E. Taylor, The Geometry of the Classical Groups, Heldermann Verlag, Berlin, 1992.
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsch. Math. Phys., vol. 3 (1892), pages 265–284.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this paper
Cite this paper
Abhyankar, S.S., Inglis, N.F.J., Yalcin, U.D. (2005). Supplementary Thoughts on Symplectic Groups. In: Voelklein, H., Shaska, T. (eds) Progress in Galois Theory. Developments in Mathematics, vol 12. Springer, Boston, MA. https://doi.org/10.1007/0-387-23534-5_1
Download citation
DOI: https://doi.org/10.1007/0-387-23534-5_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-23533-2
Online ISBN: 978-0-387-23534-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)