Tiny zero-sum sequences over some special groups

Abstract

Let S=g1⋅…⋅gnS={g}_{1}\cdot \ldots \cdot {g}_{n} be a sequence with elements gi{g}_{i} from an additive finite abelian group G. S is called a tiny zero-sum sequence if S is non-empty, g1+…+gn=0{g}_{1}+\hspace{0.2em}\ldots \hspace{0.2em}+{g}_{n}=0 and k(S)≔∑i=1n1ord(gi)≤1k(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=C3⊕C3pG={C}_{3}\oplus {C}_{3p}, where p is a prime number such that p≥5p\ge 5.

Keywords

• zero-sum sequence
• finite abelian group
• cross number
• tiny
• 05d05