The structure of modular generalized invariants

Let G be a finite group with a faithful representation V   over a field FF of odd characteristic p, where the order of G is divisible by p. In this article, it is proved that the generalized invariant module of G coincides with the (polynomial) invariant ring of the representation under a necessary and sufficient group theoretical condition. Moreover, the structure of the generalized invariant module for any finite group is determined, providing a tool for computing this module.