Long zero-free sequences in finite cyclic groups

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than n/2 in the additive group Zn of integers modulo n. The main result states that for each zero-free sequence (ai) l i=1 of length l > n/2 in Zn there is an integer g coprime to n such that if gai denotes the least positive integer in the congruence class gai (modulo n), then Σ l i=1gai < n. The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than n/2, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.