On the index of length four minimal zero-sum sequences

Caixia Shen ; Li-meng Xia ; Yuanlin Li

Colloquium Mathematicae, Tome 135 (2014), p. 201-209 / Harvested from The Polish Digital Mathematics Library

Résumé

Let G be a finite cyclic group. Every sequence S over G can be written in the form $S = (n₁g)·...·(n_{l}g)$ where g ∈ G and $n₁,..., n_{l}i ∈ [1,ord(g)]$, and the index ind(S) is defined to be the minimum of $(n₁+ ⋯ +n_{l})/ord(g)$ over all possible g ∈ G such that ⟨g⟩ = G. A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd(|G|,6) = 1 has index 1. This conjecture was confirmed recently for the case when |G| is a product of at most two prime powers. However, the general case is still open. In this paper, we make some progress towards solving the general case. We show that if G = ⟨g⟩ is a finite cyclic group of order |G| = n such that gcd(n,6) = 1 and S = (x₁g)·(x₂g)·(x₃g)·(x₄g) is a minimal zero-sum sequence over G such that x₁,...,x₄ ∈ [1,n-1] with gcd(n,x₁,x₂,x₃,x₄) = 1, and $gcd(n,x_{i}) > 1$ for some i ∈ [1,4], then ind(S) = 1. By using a new method, we give a much shorter proof to the index conjecture for the case when |G| is a product of two prime powers.

Détail

Publié le : 2014-01-01

Zbl 1301.11016

EUDML-ID : urn:eudml:doc:284209