AI 中文总结
本文通过构造二面体群D_8的反例，推翻了Wehlau关于Noether数的猜想，并展示了其在特征2的域上的普遍性。
AI 中文摘要
设G为有限群，V为有限维的G-模，U为V的G-子模。Wehlau猜想对应的Noether数满足β(k[U]^G)≤β(k[V]^G)。我们在特征2的情况下推翻了这个猜想。对于二面体群D_8，我们构造了一个包含关系U⊆V，其中dim_kU=5，dim_kV=6，使得β(k[U]^{D_8})=6>5=β(k[V]^{D_8})。该构造定义在F_2上，并且在任何特征为2的域上都成立。通过膨胀，它为每一个允许D_8作为商群的有限群提供了反例。
英文摘要
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 $β\bigl(k[U]^G\bigr)=6>5=β\bigl(k[V]^G\bigr)$, 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 $β\bigl(k[Q]^G\bigr)=8>6=β\bigl(k[Q^*]^G\bigr)$ and $β\bigl(k[Q]^G\bigr)=8>5=β\bigl(k[X]^G\bigr)$. 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.
