MINIMAL ZERO-SUM SEQUENCES OF LENGTH FIVE OVER FINITE CYCLIC GROUPS

Let G be a finite cyclic group. Every sequence S of length l over G can be written in the form S = (n(1g)) . ... . (n(lg)) where g is an element of G and n(1), ... , n(l) is an element of [1, ord(g)], and the index id(S) of S is defined to be the minimum of (n(1) + ... + n(l))/ ord(g) over all possible g is an element of G such that < g > = G. In this paper, we determine the index of any minimal zero-sum sequence S of length 5 when G = < g > is a cyclic group of a prime order and S has the form S = g(2)(n(2g))(n(3g))(n(4g)). It is shown that if G = < g > is a cyclic group of prime order p >= 31, then every minimal zero-sum sequence S of the above mentioned form has index 1 except in the case that S = g(2)(p-1/2 g)(p+3/2 g)((p - 3)g)