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

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let D ( G ) be the Davenport constant of G . Then a product of two atoms of H can be written as a product of at most D ( G ) atoms. We study this extremal case and consider the set U { 2 , D ( G ) } ( H ) defined as the set of all l ∈ N with the following property: there are two atoms u , v ∈ H such that u v can be written as a product of l atoms as well as a product of D ( G ) atoms. If G is cyclic, then U { 2 , D ( G ) } ( H ) = { 2 , D ( G ) } . If G has rank two, then we show that (apart from some exceptional cases) U { 2 , D ( G ) } ( H ) = [ 2 , D ( G ) ] ∖ { 3 } . 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 2 prime power order group uniquely characterizes the group.