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

We study the maximal cross number K(G)K(G) of a minimal zero-sum sequence and the maximal cross number k(G)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)k(G) conjectured by Krause and Zahlten holds for G⨁Cpa⨁CpbG⨁Cpa⨁Cpb 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)K(G) holds for abelian groups of the form Hp⨁CqmHp⨁Cqm (where HpHp is any finite abelian p  -group) and Cp⨁Cq⨁CrCp⨁Cq⨁Cr for any distinct primes p,q,rp,q,r. We also give a structural result on the minimal zero-sum sequences that achieve this value.