On the Olson and the Strong Davenport constants

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary $p$-groups of rank at most $2$, paralleling and building on recent results on this problem for the Olson constant

A characterization of class groups via sets of lengths

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u\_1 \cdot \ldots \cdot u\_k$, with irreducibles $u\_1, \ldots u\_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible $k$ is called the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ depends only on the class group $G$. In the present paper we study the inverse question asking whether or not the system $\mathcal L (H)$ is characteristic for the class group. Consider a further Krull monoid $H'$ with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are isomorphic (apart from two well-known pairings).Comment: The current version is close to the one to appear in J. Korean Math. Soc., yet it contains a detailed proof of Proposition 2.4. The content of Chapter 4 of the first version had been split off and is presented in ' A characterization of Krull monoids for which sets of lengths are (almost) arithmetical progressions' by the same authors (see hal-01976941 and arXiv:1901.03506

The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup

Let $G$ be a finite group written multiplicatively. By a sequence over $G$, we mean a finite sequence of terms from $G$ which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of $G$. The small Davenport constant $\mathsf d (G)$ is the maximal integer $\ell$ such that there is a sequence over $G$ of length $\ell$ which has no nontrivial, product-one subsequence. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence---this is a product-one sequence which cannot be factored into two nontrivial, product-one subsequences. It is easily observed that $\mathsf d(G)+1\leq \mathsf D(G)$, and if $G$ is abelian, then equality holds. However, for non-abelian groups, these constants can differ significantly. Now suppose $G$ has a cyclic, index 2 subgroup. Then an old result of Olson and White (dating back to 1977) implies that $\mathsf d(G)=\frac12|G|$ if $G$ is non-cyclic, and $\mathsf d(G)=|G|-1$ if $G$ is cyclic. In this paper, we determine the large Davenport constant of such groups, showing that $\mathsf D(G)=\mathsf d(G)+|G'|$, where $G'=[G,G]\leq G$ is the commutator subgroup of $G$