On the Cohen-Macaulay property of modular invariant rings

If V is a faithful module for a finite group G over a field of characteristic p, then the ring of invariants need not be Cohen-Macaulay if p divides the order of G. In this article the cohomology of G is used to study the question of Cohen-Macaulayness of the invariant ring. One of the results is a classification of all groups for which the invariant ring with respect to the regular representation is Cohen-Macaulay. Moreover, it is proved that if p divides the order of G, then the ring of vector invariants of sufficiently many copies of V is not Cohen-Macaulay. A further result is that if G is a p-group and the invariant ring is Cohen-Macaulay, then G is a bireflection group, i.e., it is generated by elements which fix a subspace of V of codimension at most 2.