On the maximal cross number of unique factorization indexed multisets

In this paper, we study a conjecture of Gao and Wang concerning a proposed formula $K_1^{\ast }(G)$ for the maximal cross number $K_1(G)$ taken over all unique factorization indexed multisets 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 $C_{p^m}\oplus C_p,C_{p^m}\oplus C_q,C_{p^m}\oplus C_q^2$, $C_{p^m}\oplus C_r^n$ where $p,q$ are distinct primes and $r\in \{2,3\}$. In our second main result we verify that $K_1(G)=K_1^{\ast }(G)$ for groups of the form $C_r\oplus C_{p^m}\oplus C_p,C_{r{p}^{m}q}$ and $C_r\oplus C_p\oplus C_q^2$ for $r\in \{2,3\}$ given some restrictions on $p$ and $q$. We also study general techniques for computing and bounding $K_1(G)$, and derive an asymptotic result which shows that $K_1(G)$ becomes arbitrarily close to $K_1^{\ast }(G)$ as the smallest prime dividing $\mathrm{\mid }G\mathrm{\mid }$ goes to infinity, given certain conditions on the structure of $G$. We also derive some necessary properties of the structure of unique factorization indexed multisets which would hypothetically violate $k(S) \le K_1^*(G)$.