Text. Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n 1 g) · . . . · (n l g) where g ∈ G and n 1 , . . . ,n l ∈ [1, ord(g)], and the index ind(S) of S is defined to be the minimum of (n 1 + · · · + n l )/ord(g) over all possible g ∈ G such that g = supp(S) . The problem regarding the index of sequences has been studied in a series of papers, and a main focus is to determine sequences of index 1. In the present paper, we show that if G is a cyclic of prime power order such that gcd(|G|, 6) = 1, then every minimal zero-sum sequence of length 4 has index 1.Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=BC7josX_xVs.
Let G be a finite cyclic group of order n ≥ 2. Every sequence S over G can be written in the form S = (n1g) · . . . · (n l g) where g ∈ G and n1, . . . , n l ∈ [1, ord(g)], and the index ind(S) of S is defined as the minimum of (n1 + . . . + n l )/ ord(g) over all g ∈ G with ord(g) = n. In this paper we prove that a sequence S over G of length |S| = n having an element with multiplicity at least n 2 has a subsequence T with ind(T ) = 1, and if the group order n is a prime, then the assumption on the multiplicity can be relaxed to n−2 10 . On the other hand, if n = 4k + 2 with k ≥ 5, we provide an example of a sequence S having length |S| > n and an element with multiplicity n 2 − 1 which has no subsequence T with ind(T ) = 1. This disproves a conjecture given twenty years ago by Lemke and Kleitman.
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