Intersections of Matrix Algebras and Permutation Representations of PSL(n, q)

If G is a group, H a subgroup of G, and Ω a transitive G-set we ask under what conditions one can guarantee that H has a regular orbit ( = of size H) on Ω. Here we prove that if PSL(n, q) G PGL(n, q) and H is cyclic then H has a regular orbit in every non-trivial G-set (with few exceptions). This result is obtained via a mixture of group theoretical and ring theoretical methods: Let R be the ring of all n × n matrices over the finite field F and let Z be the subring of scalar matrices. We show that if A and M are proper subrings of R containing Z, and if A is commutative and semisimple, then there exists an element x SL(n, F) such that xAx− 1 ∩ M = Z or n = 2 = F.