Cross Number Invariants of Finite Abelian Groups

TL;DR

This paper investigates the properties of cross number invariants in finite abelian groups, providing structural results, proving conjectures for specific group classes, and proposing new conjectures related to zero-sum sequences.

Contribution

It introduces inductive theorems for cross number invariants, proves conjectured values for certain groups, and formulates new conjectures on sequence structures in finite abelian groups.

Findings

Proved conjectured cross number values for cyclic groups with distant prime factors.

Established the $ ext{K}_1(G)$ conjecture for rank two groups of the form $C_n imes C_q$.

Formulated a conjecture on the structure of maximal-length unique factorization sequences.

Abstract

The cross number of a sequence over a finite abelian group $G$ is the sum of the inverse orders of the terms of that sequence. We study two group invariants, the maximal cross number of a zero-sum free sequence over $G$, called $\mathsf{k}(G)$, introduced by Krause, and the maximal cross number of a unique factorization sequence over $G$, called $K_{1}(G)$, introduced by Gao and Wang. Conjectured formulae for $\mathsf{k}(G)$ and $\mathsf{K}_{1}(G)$ are known, but only some special cases are proved for either. We show structural results about maximal cross number sequences that allow us to prove an inductive theorem giving conditions under which the conjectured values of $\mathsf{k}$ and $\mathsf{K}_{1}$ must be correct for $G\oplus C_{p^{\alpha}}$ if they are correct for a group $G$. As a corollary of this result we prove the conjectured values of $\mathsf{k}(G)$ and $\mathsf{K}_{1}(G)$ for cyclic groups $C_{n}$, given that the prime factors of $n$ are far apart. Our methods also prove the $\mathsf{K}_{1}(G)$ conjecture for rank two groups of the form $C_{n}\oplus C_{q}$, where $q$ is the largest or second largest prime dividing $n$, and the prime factors of $n$ are far apart, and the $\mathsf{k}(G)$ conjecture for groups of the form $C_{n}\oplus H_{q}$, where the prime factors of $n$ are far apart, $q$ is the largest prime factor of $n$, and $H_{q}$ is an arbitrary finite abelian $q$-group. Finally, we pose a conjecture about the structure of maximal-length unique factorization sequences over elementary $p$-groups, which is a major roadblock to extending the $\mathsf{K}_{1}$ conjecture to groups of higher rank, and formulate a general question about the structure of maximal zero-sum free and unique factorization sequences with respect to arbitrary weighting functions.