Counterexamples to Conjectures of Wehlau on Noether Numbers

Abstract

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 $G=D_8$, we exhibit a submodule $U\subseteq V$, with $\dim_kU=5$ and $\dim_kV=6$, such that $\beta\bigl(k[U]^G\bigr)=6>5=\beta\bigl(k[V]^G\bigr)$, 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 $\beta\bigl(k[Q]^G\bigr)=8>6=\beta\bigl(k[Q^*]^G\bigr)$ and $\beta\bigl(k[Q]^G\bigr)=8>5=\beta\bigl(k[X]^G\bigr)$. 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 $D_8$ conclusions propagate by inflation to every finite group admitting $D_8$ as a quotient.