On the index of length four minimal zero-sum sequences

Caixia Shen, Li-meng Xia, and Yuanlin Li. "On the index of length four minimal zero-sum sequences." Colloquium Mathematicae 135.2 (2014): 201-209. <http://eudml.org/doc/284209>.

@article{CaixiaShen2014,
abstract = {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.},
author = {Caixia Shen, Li-meng Xia, Yuanlin Li},
journal = {Colloquium Mathematicae},
keywords = {minimal zero-sum sequence; index of sequences},
language = {eng},
number = {2},
pages = {201-209},
title = {On the index of length four minimal zero-sum sequences},
url = {http://eudml.org/doc/284209},
volume = {135},
year = {2014},
}

TY - JOUR
AU - Caixia Shen
AU - Li-meng Xia
AU - Yuanlin Li
TI - On the index of length four minimal zero-sum sequences
JO - Colloquium Mathematicae
PY - 2014
VL - 135
IS - 2
SP - 201
EP - 209
AB - 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.
LA - eng
KW - minimal zero-sum sequence; index of sequences
UR - http://eudml.org/doc/284209
ER -