Erdős–Ginzburg–Ziv theorem and Noether number for Cm⋉φCmn

Abstract

Let G be a multiplicative finite group and S=a1⋅…⋅ak a sequence over G. We call S a product-one sequence if 1=∏i=1kaτ(i) holds for some permutation τ of {1,…,k}. The small Davenport constant d(G) is the maximal length of a product-one free sequence over G. For a subset L⊂N, let sL(G) denote the smallest l∈N0∪{∞} such that every sequence S over G of length |S|≥l has a product-one subsequence T of length |T|∈L. Denote e(G)=max⁡{ord(g):g∈G}. Some classical product-one (zero-sum) invariants including D(G):=sN(G) (when G is abelian), E(G):=s{|G|}(G), s(G):=s{e(G)}(G), η(G):=s[1,e(G)](G) and sdN(G) (d∈N) have received a lot of studies. The Noether number β(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≅Cm⋉φCmn, in this paper, we prove thatE(G)=d(G)+|G|=m2n+m+mn−2 and β(G)=d(G)+1=m+mn−1. We also prove that smnN(G)=m+2mn−2 and provide the upper bounds of η(G), s(G). Moreover, if G is a non-cyclic nilpotent group and p is the smallest prime divisor of |G|, we prove that β(G)≤|G|p+p−1 except if p=2 and G is a dicyclic group, in which case β(G)=12|G|+2.