On collineation groups of finite projective spaces containing a Singer cycle

By a result of W.M. 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 PGammaL(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)\rightarrow \PGL(n-1,q), for some divisor s of n, and so is contained in the image of GammaL(n/s,q^s) under the canonical homomorphism GammaL(n,q)\rightarrow\PGammaL(n-1,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).