Erdős-Ginzburg-Ziv theorem and Noether number for $C_m{⋉}_{\ddot{I}}†C_{mn}$

Abstract

Let $G$ be a multiplicative finite group and $S=a_1\cdot \dots \cdot a_k$ a sequence over $G$. We call $S$ a product-one sequence if $1={\prod }_{i=1}^k{a}_{\ddot{I}„(i)}$ holds for some permutation $\ddot{I}„$ of $\{1,\dots ,k\}$. The small Davenport constant $\mathsf{d}(G)$ is the maximal length of a product-one free sequence over $G$. For a subset $L\subset \mathbb{N}$, let ${\mathsf{s}}_L(G)$ denote the smallest $l\in {\mathbb{N}}_0\cup \{\infty \}$ such that every sequence $S$ over $G$ of length $|S|\ge l$ has a product-one subsequence $T$ of length $|T|\in L$. Denote $\mathsf{e}(G)=\max \{\text{ord}(g):g\in G\}$. Some classical product-one (zero-sum) invariants including $\mathsf{D}(G):={\mathsf{s}}_{\mathbb{N}}(G)$ (when $G$ is abelian), $\mathsf{E}(G):={\mathsf{s}}_{\{|G|\}}(G)$, $\mathsf{s}(G):={\mathsf{s}}_{\{\mathsf{e}(G)\}}(G)$, $\hat{I}\cdot (G):={\mathsf{s}}_{[1,\mathsf{e}(G)]}(G)$ and ${\mathsf{s}}_{d\mathbb{N}}(G)$ ($d\in \mathbb{N}$) have received a lot of studies. The Noether number ${\hat{I}}^2(G)$ which is closely related to zero-sum theory is defined to be the maximal degree bound for the generators of the algebra of polynomial invariants. Let $G\cong C_m{⋉}_{\ddot{I}}†C_{mn}$, in this paper, we prove that $$\mathsf{E}(G)=\mathsf{d}(G)+|G|=m^{2}n+m+mn-2$$ and ${\hat{I}}^2(G)=\mathsf{d}(G)+1=m+mn-1$. We also prove that ${\mathsf{s}}_{mn\mathbb{N}}(G)=m+2mn-2$ and provide the upper bounds of $\hat{I}\cdot (G)$, $\mathsf{s}(G)$. Moreover, if $G$ is a non-cyclic nilpotent group and $p$ is the smallest prime divisor of $|G|$, we prove that ${\hat{I}}^2(G)\le \frac{|G|}{p}+p-1$ except if $p=2$ and $G$ is a dicyclic group, in which case ${\hat{I}}^2(G)=\frac{1}{2}|G|+2$.

14 pages