Minimal zero-sum sequence of length five over finite cyclic groups of prime power order

2014/02/02 by Li-meng Xia, Xia, Li-meng, Yuanlin Li +3
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1402.0221

arxiv created 2014/02/02 · arxiv updated 2014/02/04

Abstract

Let G be a finite cyclic group. Every sequence S of length l over G can be written in the form S=(x₁g)⋅…⋅(x_lg) where g∈ G and x₁, …, x_l∈[1, \ord(g)], and the index \ind(S) of S is defined to be the minimum of (x₁+⋯+x_l)/\ord(g) over all possible g∈ G such that ⟨ g ⟩ =G. Recently the second and the third authors determined the index of any minimal zero-sum sequence S of length 5 over a cyclic group of a prime order where S=g²(x₂g)(x₃g)(x₄g). In this paper, we determine the index of any minimal zero-sum sequence S of length 5 over a cyclic group of a prime power order. It is shown that if G=⟨ g⟩ is a cyclic group of prime power order n=p^μ with p ≥ 7 and μ≥ 2, and S=(x₁g)(x₂g)(x₂g)(x₃g)(x₄g) with x₁=x₂ is a minimal zero-sum sequence with gcd(n,x₁,x₂,x₃,x₄,x₅)=1, then \ind(S)=2 if and only if S=(mg)(mg)(m(n-1)/(2)g)(m(n+3)/(2)g)(m(n-3)g) where m is a positive integer such that gcd(m,n)=1.

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