  We investigate the Cohen-Macaulay property for rings of invariants under
multiplicative actions of a finite group $G$. By definition, these are
$G$-actions on Laurent polynomial algebras that stabilize the multiplicative
group consisting of all monomials in the variables. For the most part, we
concentrate on the case where the base ring is the ring of rational integers.
Our main result states that if $G$ acts non-trivially and the invariant algebra
is Cohen-Macaulay then the abelianized isotropy groups $G_m/[G_m,G_m]$ of all
monomials m are generated by bireflections and at least one $G_m/[G_m,G_m]$ is
non-trivial. As an application, we prove the multiplicative version of Kemper's
3-copies conjecture.
