Abstract
By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, q s) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, q s) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, q s) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).
Similar content being viewed by others
References
Aschbacher M.: Finite group theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2000)
Baer R.L.: Linear algebra and projective geometry. Academic Press, New York (1952)
Beth, T., Jungnickel, D., Lenz, H.: Design theory. Vol. I. Second edition. Encyclopedia of Mathematics and its Applications, 69. Cambridge University Press, Cambridge (1999)
Bray, J.N., Holt, D.F., Roney-Dougal, C.: The maximal subgroups of the low-dimensional finite classical groups, London Mathematical Society Lecture Note Series, vol 407. Cambridge University Press, Cambridge (2013)
Cossidente A., De Resmini M.J.: Remarks on Singer Cyclic Groups and Their Normalizers. Design Code Cryprogr. 32, 97–102 (2004)
Dye R.H.: Spreads and classes of maximal subgroups of GL n (q), SL n (q), PGL n (q) and PSL n (q). Ann. Mat. Pura Appl. 158, 33–50 (1991)
Drudge, K.: On the orbits of Singer groups and their subgroups. Electron. J. Combin. 9, p. 10. (Electronic) (2002)
Ebert G.L., Metsch K., Szönyi T.: Caps embedded in Grassmannians. Geom. Dedicata 70, 181–196 (1998)
Glynn D.G.: On a set of lines of PG(3,q) corresponding to a maximal cap contained in the Klein quadric of PG(5,q). Geom. Dedicata 26, 273–280 (1988)
Huppert, B.: Endliche Gruppen. I, Springer-Verlag, Berlin-New York (1967)
Kantor W.M.: Linear groups containing a Singer cycle. J. Algebra 62, 232–234 (1980)
Kleidman, P., Liebeck, M.: The subgroup structure of the finite classical groups. London Mathematical Society Lecture Note Series, vol. 129. Cambridge University Press (1990)
Segre B., di Galois, Teoria: fibrazioni proiettive e geometrie non Desarguesiane. Ann. Mat. Pura Appl. (4) 64, 1–76 (1964)
Steinberg R.: Automorphisms of finite linear groups. Can. J. Math. 12, 606–615 (1960)
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author wishes to thank Tim Penttila and Mary Worthley for their kind hospitality and for many stimulating conversations.
Rights and permissions
About this article
Cite this article
Penttila, T., Siciliano, A. On collineation groups of finite projective spaces containing a Singer cycle. J. Geom. 107, 617–626 (2016). https://doi.org/10.1007/s00022-015-0300-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00022-015-0300-4