ABSTRACT
By modular invariant theory1 we understand the study of invariants of finite groups over fields of nonzero characteristic, where the characteristic may di vide the order of the group. If the group is a finite p-group and the field has characteristic p we speak of purely modular invariant theory. The invariant theory of finite groups over finite fields presents a number of special features and problems, and this chapter examines some of these. (See also chapter 10 and 11 where we pursue others.)
If q : G GL(n, F ) is a representation of a finite group whose order is di visible by the characteristic of F then the averaging operator derived from the transfer (see section 2.4) is no longer defined. This loss is partially compen sated by the fact that for a Galois field F the general linear group GL(n, F) is a finite group, and the ring of invariants F [V ]GL(v ) is nontrivial and con tained in W[V]G for any g : G GL(n, F ), where as usual V = F n. Thus there exist universal invariants, i.e. invariants present in the ring of invariants of all finite group over a finite field.
