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)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⊕CpCpm⊕Cp, Cpm⊕CqCpm⊕Cq, Cpm⊕Cq2, Cpm⊕Crn where p, q   are distinct primes and r∈{2,3}r∈{2,3}. In our second main result we verify that K1(G)=K1⁎(G) for groups of the form Cr⊕Cpm⊕CpCr⊕Cpm⊕Cp, CrpmqCrpmq and Cr⊕Cp⊕Cq2 for r∈{2,3}r∈{2,3} given some restrictions on p and q  . We also study general techniques for computing and bounding K1(G)K1(G), and derive an asymptotic result which shows that K1(G)K1(G) becomes arbitrarily close to K1⁎(G) as the smallest prime dividing |G||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).