On collineation groups of finite projective spaces containing a Singer cycle

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 PGL(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 Gamma L(n/s, q(s)) under the canonical homomorphism Gamma L(n, q) -> P -> L(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).