On the index of length four minimal zero-sum sequences

• Autorzy: Caixia Shen , Li-meng Xia , Yuanlin Li
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2014
• Tom: 135
• Numer: 2
• Strony: 201-209

Daty

wydano

2014

Twórcy

autor

Caixia Shen

• Faculty of Science, Jiangsu University, Zhenjiang, 212013, Jiangsu Prov., China

autor

Li-meng Xia

• Faculty of Science, Jiangsu University, Zhenjiang, 212013, Jiangsu Prov., China

autor

Yuanlin Li

• Department of Mathematics, Brock University, St. Catharines, ON, Canada L2S 3A1

Identyfikatory

DOI

10.4064/cm135-2-4