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) ≤

Let G be an additive finite abelian group with exponent exp(G). Let η(G) be the smallest integer t such that every sequence of length t has a nonempty zero-sum subsequence of length at most exp(G). Let s(G) be the EGZ-constant of G, which is defined as the smallest integer t such that every sequence of length t has a zero-sum subsequence of length exp(G).

Let [Formula: see text] be an additive finite abelian group with exponent [Formula: see text]. For any positive integer [Formula: see text], let [Formula: see text] be the smallest positive integer [Formula: see text] such that every sequence [Formula: see text] in [Formula: see text] of length at least [Formula: see text] has a zero-sum subsequence of length [Formula: see text]. Let [Formula: see text] be the Davenport constant of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite abelian [Formula: see text]-group with [Formula: see text] then [Formula: see text] for every [Formula: see text], which confirms a conjecture by Gao et al. recently, where [Formula: see text] is a prime.

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