Cross numbers and the characterization of finite abelian groups by length sets

Abstract

The Characterization Problem asks whether a finite abelian group with Davenport constant at least four is determined, up to isomorphism, by its system of length sets. This is a central inverse problem in non-unique factorization theory: it asks whether the possible numbers of irreducible factors retain enough information to recover the underlying class group. For rings of integers of algebraic number fields, an affirmative answer means that the ideal class group can be recovered from these factorization lengths alone. We resolve this problem for all homocyclic groups $C_n^r$: if $G$ is a finite abelian group with $\mathsf D(G)\ge4$ and $\mathcal L(G)=\mathcal L(C_n^r)$, then $G\cong C_n^r$. The cross number of a sequence is the sum of the reciprocals of the orders of its terms. This weighted measure of sequence length links zero-sum theory with half-factoriality, and determining its maximal value for minimal zero-sum sequences is a longstanding extremal problem. We also obtain an exact formula for the maximal cross number of minimal zero-sum sequences over subsets whose minimum distance is divisible by the rank of the ambient group minus one. For $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ with $1<n_1\mid\cdots\mid n_r$ and $r\ge\max\{2,n_r-2\}$, we prove \[ \mathsf K(G,r-1)=1+(r-1)\sum_{p\mid n_1} \left(1-p^{-\mathsf v_p(n_1)}\right), \] which is a key ingredient in resolving the Characterization Problem for homocyclic groups.