On a conjecture concerning the maximal cross number of unique factorization indexed sequences

In this paper, we study a conjecture of Gao and Wang concerning a proposed formula K1*(G) for the maximal cross number K1(G) taken over all unique factorization indexed sequences over a given finite abelian group G. As a corollary of our first main result, we verify the conjecture for abelian groups of the form Cpm⊕Cp, Cpm⊕Cq, Cpm⊕Cq2, Cpm⊕Crn where p, q are distinct primes and r ∈ {2, 3}. In our second main result we verify that K1(G)=K1*(G) for groups of the form Cr⊕Cpm⊕Cp, Crpmq and Cr⊕Cp⊕Cq2 for r ∈ {2, 3} given some restrictions on p and q. We also study general techniques for computing and bounding K1(G), and derive an asymptotic result which shows that K1(G) becomes arbitrarily close to K1*(G) as the smallest prime dividing |G| goes to infinity, given certain conditions on the structure of G. We also derive some results on the structure of unique factorization indexed sequences which would hypothetically violate k-(S)≤K1*(G).