Products of two atoms in Krull monoids and arithmetical characterizations of class groups

Let [Formula: see text] be a Krull monoid with finite class group [Formula: see text] such that every class contains a prime divisor and let [Formula: see text] be the Davenport constant of [Formula: see text]. Then a product of two atoms of [Formula: see text] can be written as a product of at most [Formula: see text] atoms. We study this extremal case and consider the set [Formula: see text] defined as the set of all [Formula: see text] with the following property: there are two atoms [Formula: see text] such that [Formula: see text] can be written as a product of [Formula: see text] atoms as well as a product of [Formula: see text] atoms. If [Formula: see text] is cyclic, then [Formula: see text]. If [Formula: see text] has rank two, then we show that (apart from some exceptional cases) [Formula: see text]. This result is based on the recent characterization of all minimal zero-sum sequences of maximal length over groups of rank two. As a consequence, we are able to show that the arithmetical factorization properties encoded in the sets of lengths of a rank [Formula: see text] prime power order group uniquely characterizes the group.