A new upper bound for the cross number of finite Abelian groups

TL;DR

This paper establishes a new upper bound for the cross number of finite Abelian groups, depending on their rank and prime divisors, advancing understanding of their combinatorial properties.

Contribution

It introduces a novel upper bound for the little cross number applicable to all finite Abelian groups, extending previous cyclic group results.

Findings

New upper bound depends on rank and prime divisors

Proves asymptotic validity of a classical conjecture in multiple cases

Enhances understanding of cross number behavior in finite Abelian groups

Abstract

In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite Abelian groups. Given a finite Abelian group, this upper bound appears to depend only on the rank and on the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite Abelian groups holds asymptotically in at least two different directions.