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

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 $a=u_1\cdot \dots \cdot u_k$ into irreducible elements $u_1,\dots ,u_k\in H$, 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 $H^{\prime }$ be a further Krull monoid with class group $G^{\prime }$ such that every class contains a prime divisor and suppose that $\mathcal{L}(H)=\mathcal{L}(H^{\prime })$. We prove that, if one of the groups $G$ and $G^{\prime }$ is isomorphic to $C_n^r$ with $r,n$ as above, then $G$ and $G^{\prime }$ are isomorphic (apart from two well-known pairings).

References in corpus (3)

• A characterization of class groups via sets of lengths
• On products of k atoms II
• On congruence half-factorial Krull monoids with cyclic class group

Cited by in corpus (8)

• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids
• Long sets of lengths with maximal elasticity
• Factorization invariants in numerical monoids
• On products of k atoms II
• On congruence half-factorial Krull monoids with cyclic class group
• On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II
• Factorization Theory in Commutative Monoids
• A characterization of finite abelian groups via sets of lengths in transfer Krull monoids