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Let k be a field of positive characteristic p and let G be a finite group. Inthis paper we study the category TsG of finitely generated commutativek-algebras A on which G acts by algebra automorphisms with surjective trace. IfA = k[X], the ring of regular functions of a variety X, then trace-surjectivegroup actions on A are characterized geometrically by the fact that all pointstabilizers on X are p'-subgroups or, equivalently, that A is a Galoisextension of the ring of P invariants for every Sylow p-group P of G. Weinvestigate categorical properties, using a version of Frobenius-reciprocityfor group actions on k-algebras, which is based on tensor induction formodules. We also describe projective generators in TsG , extending andgeneralizing the investigations started in [8], [7] and [9] in the case ofp-groups. As an application we show that for an abelian or p-elementary group Gand k large enough, there is always a faithful (possibly nonlinear) action on apolynomial ring such that the ring of invariants is also a polynomial ring.This would be false for linear group actions by a result of Serre. Forp-solvable groups we obtain a structure theorem on trace-surjective algebras,generalizing the corresponding result for p-groups in [8].

Modular group actions on algebras and p-local Galois extensions for finite groups