P*-invariant ideals in rings of invariants

In this article we study modular invariants of finite groups using as tools, the Steenrod algebra and the Dickson algebra. The ring of invariants of a finite group over the field F-p of p elements is an unstable algebra over the Steenrod algebra. We extend this to arbitrary Galois fields and exploit this extra structure to study the transfer map Tr(G) : F[V] --> F[V](G). The case G = GL(n, F-q) is a universal example in the sense that classes in Im(Tr(GL(n, Fq))) lie in Im(Tr(G)) for any rho : G hooked right arrow GL(n, F-q). We will show that the radical of the ideal Im(Tr(GL(n, Fq))) is the principal ideal generated by the top Dickson class d(n, 0), a result first proved by M. Feshbach for the prime 2.