Invariant theory of finite sparse linear groups

In the nonmodular case finite groups whose rings of invariants are polynomial algebras are characterized by Chevalley-Shephard-Todd as pseudoreflection groups, while in modular cases no complete characterization is known. Motivated by the phenomenon that rings of invariants of GL(n,F_q) and various of its subgroups are polynomial, we construct a large family of subgroups in GL(n,F_q) and prove that they all have polynomial rings of invariants. This construction includes many known examples as well as all p-groups with polynomial invariants over F_p.