A characterization of class groups via sets of lengths

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a\in H$ can be written as a finite product of irreducible elements. If $a=u\mathrm{\_}1\cdot \dots \cdot u\mathrm{\_}k$, with irreducibles $u\mathrm{\_}1,\dots u\mathrm{\_}k\in H$, then $k$ is called the length of the factorization and the set $\mathsf{L}(a)$ of all possible $k$ is called the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = {\mathsf L (a) \mid a \in H }$dependsonlyontheclassgroup$G$. In the present paper we study the inverse question asking whether or not the system $\mathcal L (H)$ischaracteristicfortheclassgroup\mathrm{.}ConsiderafurtherKrullmonoid$H’$withclassgroup$G’$ such that every class contains a prime divisor and suppose that $\mathcal{L}(H)=\mathcal{L}(H^{\mathrm{\prime }})$. We show that, if one of the groups $G$ and $G^{\mathrm{\prime }}$ is finite and has rank at most two, then $G$ and $G^{\mathrm{\prime }}$ are isomorphic (apart from two well-known pairings).