A characterization of linearly reductive groups by their invariants

The theorem of Hochster and Roberts says that, for every module V of a line arly reductive group G over a field K, the invariant ring K[V](G) is Cohen- Macaulay. We prove the following converse: if G is a reductive group and K[ V](G) is Cohen-Macaulay for every module V, then G is linearly reductive.