Erdos-Ginzburg-Ziv theorem and Noether number for Cm ∝φ Cmn

Let G be a multiplicative finite group and S = a(1) ..... a(k) a sequence over G. We call S a product-one sequence if 1 = Pi(k)(i=1) a(tau(i)) holds for some permutation tau 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 subset of N, let s(L)(G) denote the smallest l is an element of N-0 U {infinity} such that every sequence S over G of length vertical bar S vertical bar >= l has a product-one subsequence T of length vertical bar T vertical bar is an element of L. Denote e(G) = max{ord(g) : g is an element of G}. Some classical product-one (zero-sum) invariants including D(G) := s(N)(G) (when G is abelian), E(G) := s({vertical bar G vertical bar})(G), s(G) := S ({e(G)})(G), eta(G) := s([1,e(G)]) (G) and s(dN)(G) (d is an element of N) have received a lot of studies. The Noether number beta(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 congruent to C-m proportional to(phi) C-mn, in this paper, we prove that E(G) = d(G) + vertical bar G vertical bar = m(2)n + m + mn - 2 and beta(G) = d(G) + 1 = m + mn - 1. We also prove that s(mnN)(G) = m + 2mn - 2 and provide the upper bounds of eta(G), s(G). Moreover, if G is a non-cyclic nilpotent group and p is the smallest prime divisor of vertical bar G vertical bar, we prove that beta(G) <= vertical bar G vertical bar/p + p - 1 except if p = 2 and G is a dicyclic group, in which case beta(G) = 1/2 vertical bar G vertical bar + 2. (C) 2018 Elsevier Inc. All rights reserved.