Non-Cohen–Macaulay invariant rings of infinite groups

We give explicit examples of invariant rings that are not Cohen–Macaulay for all classical groups SLn(K), GLn(K), Sp2n(K), SOn(K) and On(K), where K is an algebraically closed field of positive characteristic. We prove that every non-trivial unipotent group over K has representations such that the invariant ring is not Cohen–Macaulay.