  Let $G$ be a finite group acting linearly on the vector space $V$ over a
field of arbitrary characteristic. The action is called {\em coregular} if the
invariant ring is generated by algebraically independent homogeneous invariants
and the {\em direct summand property} holds if there is a surjective
$k[V]^G$-linear map $\pi:k[V]\to k[V]^G$. The following
Chevalley--Shephard--Todd type theorem is proved. Suppose $G$ is abelian, then
the action is coregular if and only if $G$ is generated by pseudo-reflections
and the direct summand property holds.
