p-Groups and the polynomial ring of invariants question

Let G⊂GL(V) be a finite p-group, where p=charF and dim_F⁡V is finite. Let S(V) be the symmetric algebra of V, S(V)^G the subring of G-invariants, and V^⁎ the dual space of V. We determine when S(V)^G is a polynomial ring. Theorem A Suppose dim_F⁡V=3. Then S(V)^G is a polynomial ring if and only if G is generated by transvections. Theorem B Suppose dim_F⁡V⩾4. Then S(V)^G is a polynomial ring if and only if: (1) S(V)^G_U is a polynomial ring for each subspace U⊂V^⁎ with dim_F⁡U=2, where G_U={g∈G|g(u)=u, ∀u∈U}, and (2) S(V)^G is Cohen-Macaulay. Alternatively, (1) can be replaced by the equivalent condition: (3) dim⁡(non-smooth locus ofS(V)^G)⩽1.