  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 $a=u\_1 \cdot \ldots \cdot u\_k$, with
irreducibles $u\_1, \ldots u\_k \in H$, then $k$ is called the length of the
factorization and the set $\mathsf L (a)$ of all possible $k$ is called the set
of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf
L (a) \mid a \in H \}$ depends only on the class group $G$. In the present
paper we study the inverse question asking whether or not the system $\mathcal
L (H)$ is characteristic for the class group. Consider a further Krull monoid
$H'$ with class group $G'$ such that every class contains a prime divisor and
suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the
groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are
isomorphic (apart from two well-known pairings).
