Back to Search
Start Over
On collineation groups of finite projective spaces containing a Singer cycle
- Source :
- Journal of Geometry. 107:617-626
- Publication Year :
- 2015
- Publisher :
- Springer Science and Business Media LLC, 2015.
-
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, qs) 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, qs) 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, qs) 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).
Details
- ISSN :
- 14208997 and 00472468
- Volume :
- 107
- Database :
- OpenAIRE
- Journal :
- Journal of Geometry
- Accession number :
- edsair.doi...........6ded31c99cf1acc9f9edfb7a21a780e9