2026/07/20 by Muhammad Fazeel Anwar
#math.RT #math.AC #math.RA
Let G be a finite group and let V be a finite-dimensional G-module over a field k. We construct explicit counterexamples in characteristic 2 to several questions and conjectures of Wehlau concerning Noether numbers. For the 2-group G=D8, we exhibit a submodule U⊆ V, with dimkU=5 and dimkV=6, such that β(k[U]G)=6>5=β(k[V]G), thereby disproving submodule monotonicity. Writing X=V^* and Q=U^*, the corresponding nonsplit exact sequence 0\longrightarrow k\longrightarrow X\longrightarrow Q\longrightarrow0 also satisfies β(k[Q]G)=8>6=β(k[Q^*]G) and β(k[Q]G)=8>5=β(k[X]G). Thus the modular Noether number need not be invariant under duality, and quotient monotonicity also fails. Notably, the basic counterexamples already occur for 2-groups in defining characteristic. The constructions remain valid over every field of characteristic 2, and the D8 conclusions propagate by inflation to every finite group admitting D8 as a quotient.