ON THE MAXIMAL CROSS NUMBER OF UNIQUE FACTORIZATION ZERO-SUM SEQUENCES OVER A FINITE ABELIAN GROUP

摘要

Let S = (g1, · · · , gl) be a sequence of elements from an additive finite abelian group G, and let k(S) = ! l i=1 1 ord(gi) denote the cross number of S. A zero-sum sequence S of nonzero elements from G is called a unique factorization sequence if S can be written in the form S = S1 · · · Sr uniquely, where all Si are minimal zero-sum subsequences of S. In this short note we investigate the following invariant of G concerning the cross number of unique factorization sequences. Define K1(G) = max{k(S)|S is a unique factorization sequence over G {0}}, where the maximum is taken when S runs over all unique factorization sequences over G {0}. We determine K1(G) for some special groups including the cyclic groups of prime power order.