A Characterization of class groups via sets of lengths {II}

TL;DR

This paper proves that for certain finite abelian class groups, the system of sets of lengths uniquely determines the group up to isomorphism, confirming a conjecture in factorization theory.

Contribution

It verifies the conjecture that the system of sets of lengths characterizes the class group for groups isomorphic to $C_n^r$ with specified parameters.

Findings

The system of sets of lengths determines the class group for groups $C_n^r$ with $r,n ext{ as specified}$.

Two non-isomorphic groups with the same system of lengths are only known in two specific pairings.

The result confirms the conjecture for a broad class of finite abelian groups.

Abstract

Let $H$ be a Krull monoid with finite class group $G$ and suppose that every class contains a prime divisor. If an element $a \in H$ has a factorization $a=u_1 \cdot \ldots \cdot u_k$ into irreducible elements $u_1, \ldots, u_k \in H$, then $k$ is called the length of the factorization and 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 class group $G$, and a standing conjecture states that conversely the system $\mathcal L (H)$ is characteristic for the class group. We verify the conjecture if the class group is isomorphic to $C_n^r$ with $r,n \ge 2$ and $r \le \max \{2, (n+2)/6\}$. Indeed, let $H'$ be a further Krull monoid with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H)= \mathcal L (H')$. We prove that, if one of the groups $G$ and $G'$ is isomorphic to $C_n^r$ with $r,n$ as above, then $G$ and $G'$ are isomorphic (apart from two well-known pairings).