Indexes of long zero-sum sequences over cyclic groups

Let G be a cyclic group of order n , and let S ∈ F ( G ) be a zero-sum sequence of length | S | ≥ 2 ⌊ n / 2 ⌋ + 2 . Suppose that S can be decomposed into a product of at most two minimal zero-sum sequences. Then there exists some g ∈ G such that S = ( n 1 g ) ⋅ ( n 2 g ) ⋅ ⋯ ⋅ ( n | S | g ) , where n i ∈ [ 1 , n ] for all i ∈ [ 1 , | S | ] and n 1 + n 2 + ⋯ + n | S | = 2 n . And we also generalize the above result to long zero-sum sequences which can be decomposed into at most k ≥ 3 minimal zero-sum sequences.