The Gorenstein property for modular binary forms invariants

Let G⊆SL(2, F) be a finite group, V=F² the natural SL(2, F)-module, and charF=p>0. Let S(V) be the symmetric algebra of V and S(V)^G the ring of G-invariants. We provide examples of groups G, where S(V)^G is Cohen-Macaulay, but is not Gorenstein. This refutes a natural conjecture due to Kemper, Körding, Malle, Matzat, Vogel and Wiese. Let T(G) denote the subgroup generated by all transvections of G. We show that S(V)^G is Gorenstein if and only if one of the following cases holds:(1)T(G)={1_G},(2)V is an irreducible T(G)-module,(3)V is a reducible T(G)-module and |G| divides |T(G)|(|T(G)|-1).