A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

Let H be a transfer Krull monoid over a finite abelian group G (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit a∈H can be written as a product of irreducible elements, say [Image: see text] , and the number of factors k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system ℒ(H) = {L(a)∣a∈H} of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system ℒ(H) is characteristic for the group G. Let H (′) be a further transfer Krull monoid over a finite abelian group G (′) and suppose that ℒ(H) = ℒ(H (′)). We prove that, if [Image: see text] with r≤n−3 or (r≥n−1≥2 and n is a prime power), then G and G (′) are isomorphic.