Minimal zero sum sequences of length four over finite cyclic groups (Q983302)

Let \(C_n\) be an additive cyclic group of order \(n\). Let \(S=g_1 \dots g_k\) be a minimal zero-sum sequence over \(C_n\), i.e., \(\sum_{i=1}^k g_i = 0\) yet \(\sum_{i \in I} g_i \neq 0\) for every proper and non-empty subset of \(\{1, \dots, k\}\). The sequence is said to have index \(1\) if there exist a generating element \(e\) of \(C_n\) and \(1 \leq a_i \leq n \) such that \(g_i = a_i e\) for each \(i\) and \(\sum_{i=1}^k a_i = n\). In view of various earlier results on minimal zero-sum sequences of index \(1\) the following question is relevant: Suppose \(n\) is co-prime to \(6\) and \(k=4\). Does it follow that the index of \(S\) is \(1\) ? In the paper under review a partial answer is given. Namely, it is proved that if \(n\) is a prime power co-prime to \(6\) and \(k=4\), then it follow that the index of \(S\) is \(1\).