The cross number of minimal zero-sum sequences in finite abelian groups

We study the maximal cross number K(G) of a minimal zero-sum sequence and the maximal cross number k(G) of a zero-sum free sequence over a finite abelian group G, defined by Krause and Zahlten.. In the first part of this paper, we extend a previous result by X. He to prove that the value of k(G) conjectured by Krause and Zahlten holds for G circle plus C-pa circle plus C-pb when it holds for G, provided that p and the exponent of G are related in a specific sense. In the second part, we describe a new method for proving that the conjectured value of K(G) holds for abelian groups of the form H-p circle plus C-qm (where H-p is any finite abelian p-group) and C-p circle plus C-q circle plus C-r for any distinct primes p, q, r. We also give a structural result on the minimal zero-sum sequences that achieve this value. (C) 2015 Elsevier Inc. All rights reserved.