Abstract
Abstract Let S = g 1 ⋅ … ⋅ g n S={g}_{1}\cdot \ldots \cdot {g}_{n} be a sequence with elements g i {g}_{i} from an additive finite abelian group G. S is called a tiny zero-sum sequence if S is non-empty, g 1 + … + g n = 0 {g}_{1}+\hspace{0.2em}\ldots \hspace{0.2em}+{g}_{n}=0 and k ( S ) ≔ ∑ i = 1 n 1 ord ( g i ) ≤ 1 k(S):= {\sum }_{i=1}^{n}\frac{1}{\text{ord}({g}_{i})}\le 1 . Let t ( G ) t(G) be the smallest integer t such that every sequence of t elements (repetition allowed) from G contains a tiny zero-sum sequence. In this article, we mainly focus on the explicit value of t ( G ) t(G) and compute this value for a new class of groups, namely ones of the form G = C 3 ⊕ C 3 p G={C}_{3}\oplus {C}_{3p} , where p is a prime number such that p ≥ 5 p\ge 5 .
Highlights
Let G be an additively written finite abelian group with exp(G) its exponent and let (G) be the free abelian monoid, multiplicatively written, with basis G
A sequence T is a subsequence of S if vg(T) ≤ vg(S) for every g ∈ G, denoted by T|S
By σ(S) we denote the sum of S, that is, σ(S) = ∑li=1 gi ∈ G
Summary
Let G be an additively written finite abelian group with exp(G) its exponent and let (G) be the free abelian monoid, multiplicatively written, with basis G. Let η(G) denote the smallest integer t ∈ such that every sequence S over G of length |S| ≥ t contains a non-empty zero-sum subsequence S′|S with |S′| ≤ exp(G). Such a subsequence is called a short zero-sum subsequence. For all finite abelian groups of rank 2, Girard [1] conjectured that t(G) = η(G) and proved that t(Cp2α) = η(Cp2α) = 3pα − 2 for any prime p.
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