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Let 
$G$ be a finite group acting linearly on the vector space 
$V$ over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants, and the 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.
