A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

TL;DR

This paper investigates how the sets of lengths of factorizations in transfer Krull monoids over finite abelian groups uniquely characterize the group, proving this for specific classes of groups.

Contribution

It proves that for certain finite abelian groups, the system of sets of lengths uniquely determines the group up to isomorphism.

Findings

Sets of lengths depend only on the group G.

Under specified conditions, the system of lengths characterizes G uniquely.

The result applies to groups of the form C_n^r with r ≤ n-3 or r ≥ n-1 prime power.

Abstract

Let $H$ be a transfer Krull monoid over a finite ablian group $G$ (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit $a \in H$ can be written as a product of irreducible elements, say $a = u_1 \ldots u_k$, and the number of factors $k$ is called the length of the factorization. The set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. It is classical that the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths depends only on the group $G$, and a standing conjecture states that conversely the system $\mathcal L (H)$ is characteristic for the group $G$. Let $H'$ be a further transfer Krull monoid over a finite ablian group $G'$ and suppose that $\mathcal L (H)= \mathcal L (H')$. We prove that, if $G\cong C_n^r$ with $r\le n-3$ or ($r\ge n-1\ge 2$ and $n$ is a prime power), then $G$ and $G'$ are isomorphic.