  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 that 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 separating algebra is
Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a
Cohen-Macaulay graded separating algebra implies the group is generated by
bireflections. Furthermore, we show that, for a $p$-group, the existence of a
Cohen-Macaulay graded separating algebra implies the group is generated by
bireflections. Additionally, we give an example which shows that Cohen-Macaulay
separating algebras can occur when the ring of invariants is not
Cohen-Macaulay.
