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

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 $u_1, \ldots, u_k \in H$,then$k$iscalledthelengthofthefactorizationandtheset$\mathsf L (a)$ofallpossiblefactorizationlengthsisthesetoflengthsof$a$. 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 $r\le \max \{2,(n+2)\mathrm{/}6\}$. Indeed, let $H^{\mathrm{\prime }}$ be a further Krull monoid with class group $G^{\mathrm{\prime }}$ such that every class contains a prime divisor and suppose that $\mathcal{L}(H)=\mathcal{L}(H^{\mathrm{\prime }})$. We prove that, if one of the groups $G$ and $G^{\mathrm{\prime }}$ is isomorphic to $C_n^r$ with $r,n$ as above, then $G$ and $G^{\mathrm{\prime }}$ are isomorphic (apart from two well-known pairings).