On the maximal cross number of unique factorization indexed multisets

TL;DR

This paper investigates a conjecture about the maximal cross number of unique factorization multisets over finite abelian groups, verifying it for specific group structures and exploring asymptotic behavior.

Contribution

It verifies Gao and Wang's conjecture for certain classes of finite abelian groups and studies the asymptotic relationship between the actual and proposed maximal cross numbers.

Findings

Confirmed the conjecture for groups like C_{p^m}⊕C_p and others.

Proved that K_1(G) equals K_1^*(G) for specific group forms under certain conditions.

Showed that K_1(G) approaches K_1^*(G) as the smallest prime dividing |G| increases.

Abstract

In this paper, we study a conjecture of Gao and Wang concerning a proposed formula $K_1^*(G)$ 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 $K_1(G) = K_1^*(G)$ for groups of the form $C_r\oplus C_{p^m}\oplus C_p, C_{rp^mq}$ 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 $K_1^*(G)$ as the smallest prime dividing $|G|$ 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 $k(S) \le K_1^*(G)$.