The Gorenstein property for modular binary forms invariants

Let G⊆SL(2,F)G⊆SL(2,F) be a finite group, V=F2V=F2 the natural SL(2,F)SL(2,F)-module, and charF=p>0charF=p>0. Let S(V)S(V) be the symmetric algebra of V   and S(V)GS(V)G the ring of G-invariants. We provide examples of groups G  , where S(V)GS(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)T(G) denote the subgroup generated by all transvections of G  . We show that S(V)GS(V)G is Gorenstein if and only if one of the following cases holds:(1)T(G)={1G}T(G)={1G},(2)V   is an irreducible T(G)T(G)-module,(3)V   is a reducible T(G)T(G)-module and |G||G| divides |T(G)|(|T(G)|−1)|T(G)|(|T(G)|−1).