The Cohen-Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is
a subalgebra of the ring of invariants which separates the orbits.
Although separating algebras are often better behaved than the
ring of invariants, we show that many of the criteria which imply
the ring of invariants is non Cohen-Macaulay actually imply that
no graded separating algebra is Cohen-Macaulay. For example, we
show that, over a field of positive characteristic p, given sufficiently
many copies of a faithful modular representation, no graded sep-
arating algebra is Cohen-Macaulay. Furthermore, we show that,
for a p-group, the existence of a Cohen-Macaulay graded separat-
ing algebra implies the group is generated by bireflections. Ad-
ditionally, we give an example which shows that Cohen-Macaulay
separating algebras can occur when the ring of invariants is not
Cohen-Macaulay.