MINIMAL ZERO-SUM SEQUENCE OF LENGTH FIVE OVER FINITE CYCLIC GROUPS OF PRIME POWER ORDER

Let G be a finite cyclic group. Every sequence S of length l over G can be written in the form S = (x1g) · ... · (xlg) where g ∈ G and x1,...,xl ∈ (1,ord(g)), and the index ind(S) of S is defined to be the minimum of (x1 + ··· + xl)/ord(g) over all possible g ∈ G such that hgi = 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 = g 2 (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 = hgi 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.