Erdős–Ginzburg–Ziv theorem and Noether number for C⋉C

Abstract

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 = k i=1 a τ (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 subsethave 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 ⋉ϕ Cmn, in this paper, we prove that E(G) = d(G) + |G| = m 2 n + m + mn − 2 and β(G) = d(G) + 1 = m + mn − 1. We also prove that s mnN (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) ≤

On a conjecture of Zhuang and Gao

“…Recently, in 2015 Han [16] verified the conjecture for the case when G is a non-cyclic nilpotent group or when G ∼ = C p ⋉ C pn , where p is a prime. Most recently, in 2019 Han and Zhang [17] We remark that the investigation of the above mentioned class of metacyclic groups is of special interest as it is closely related to solving Conjecture 1.2. Very recently, Gao, Li, and Qu [10] made substantial progress on this conjecture and proved that E(G) ≤ 3|G| 2 holds for all non-cyclic groups of odd order by using the minimal counterexample method.…”

A reciprocity on finite abelian groups involving zero-sum sequences II

“…This paper provides some results regarding zero-sum sequences over finite abelian groups and polynomial invariants of finite groups. Recently, the relationship between zero-sum theory (also factorization theory) and invariant theory is getting closer; see [2,3,4,14] for some recent studies. Based on Theorems 1.1, 1.2, and 1.3, it is natural to propose the following conjecture.…”

On the invariant $\mathsf E(G)$ for groups of odd order

“…As a striking example, we mention that the entire Chapter 16 of Grynkiewicz's monograph [13] is devoted to a generalization towards weighted zero-sum sequences (over abelian groups). This relation also holds for some non-abelian groups such as dihedral or dicyclic groups (see [1,10,15,16,22] for details). Recently in [19] the above relation has been verified for some special metacyclic groups C p C m (which are not necessarily dihedral groups).…”

“…Proof. For the groups in this lemma, proofs of E(G) ≤ |G| + |G|/p + p − 2 can be found in [15,16,19], where p is the smallest prime divisor of G. Since G is of odd order and…”