## Systems of sets of lengths. II.(English)Zbl 1036.11054

Let $$G$$ be a finite additive abelian group of exponent $$n$$. For a subset $$G_0 \subset G$$, let $$\mathcal B (G_0)$$ be the set of all (finite) sequences in $$G_0$$ which sum to zero. For $$B \in \mathcal B (G_0)$$, let $$L(B)$$ be the set of all $$r \in \mathbb N$$ such that $$B = B_1 \cdot\ldots\cdot B_r$$ with irreducible $$B_i \in \mathcal B(G_0)$$. For a finite set $$L = \{a_1, \ldots, a_k\} \subset \mathbb Z$$ with $$a_1 < a_2 < \ldots < a_k$$, set $$\Delta (L) = \{a_i - a_{i-1} \mid 2 \leq i \leq k\}$$, and in particular $$\Delta (L) = \emptyset$$ if $$k = 1$$. Let $$\Delta (G_0)$$ be the union of all sets $$\Delta (L(B))$$, where $$B \in \mathcal B (G_0)$$, and $\Delta^* (G) = \{ \min \Delta(G_0) \mid G_0 \subset G, \;\Delta (G_0) \neq \emptyset \} \,.$ $$\Delta^* (G)$$ is a key invariant in the study of non-unique factorizations in Krull monoids with finite class group (and thus in rings of integers in finite algebraic number fields); see the survey articles in [D. D. Anderson (ed.), Factorization in integral domains, Lecture Notes in Pure and Applied Mathematics 189, Marcel Dekker, New York (1997; Zbl 0865.00039)].
The authors prove the following results: 1. If $$| G| \leq n^2$$ or $$G$$ is a $$p$$-group with “small” rank, then $$\max \Delta^* (G) = n-2$$; 2. If $$G$$ is cyclic, then $$\max (\Delta^* (G) \setminus \{n-2\}) < \frac{n+1}{2}$$ (the smaller bound $$\lfloor \frac{n}{2} \rfloor -1$$ was obtained recently by A. Geroldinger and Y. O. Hamidoune [J. Théor. Nombres Bordx. 14, 221–239 (2002; Zbl 1018.11011)]); 3. If $$G$$ is a $$p$$-group of “large” rank $$r$$, then $$\Delta^*(G) = \{1,2,\ldots, r-1\}$$.
The proofs use subtle information concerning the infrastructure of zero-sum sequences and relations with various other invariants studied in additive group theory.

### MSC:

 11R27 Units and factorization 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 20D60 Arithmetic and combinatorial problems involving abstract finite groups

### Citations:

Zbl 0882.13027; Zbl 0903.13008; Zbl 0865.00039; Zbl 1018.11011
Full Text:

### References:

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