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Let G ∼ = Cn 1 ⊕ · · · ⊕ Cn r be a finite and nontrivial abelian group with n 1 |n 2 |. .. |nr. A conjecture of Hamidoune says that if W = w 1 · · · wn is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S| ≥ |W | + |G| − 1 ≥ |G| + 1, the maximum multiplicity of S at most |W |, and σ(W) ≡ 0 mod |G|, then there exists a nontrivial subgroup H such that every element g ∈ H can be represented as a weighted subsequence sum of the form g = n P i=1 w i s i , with s 1 · · · sn a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W | ≥ 1 2 |G|. A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S| ≥ |G| + D(G) − 1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g + H such that all but at most |G/H| − 2 terms of S are from g + H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S| ≥ |G| + D(G) − 1 can be relaxed to |S| ≥ |G| + d * (G), where d * (G) = r P i=1 (n i − 1). We also use this method to derive a variation on Hamidoune's conjecture valid when at least d * (G) of the w i are relatively prime to |G|. 1. Notation We follow the conventions of [9] for notation concerning sequences over an abelian group. For real numbers a, b ∈ R, we set [a, b] = {x ∈ Z | a ≤ x ≤ b}. Throughout, all abelian groups will be written additively. Let G be an abelian group, and let A, B ⊆ G be nonempty subsets. Then A + B = {a + b | a ∈ A, b ∈ B} denotes their sumset. The stabilizer of A is defined as H(A) = {g ∈ G | g + A = A}, and A is called periodic if H(A) = {0}, and aperiodic otherwise. If A is a union of H-cosets (i.e., H ≤ H(A)), then we say A is H-periodic. The order of an element g ∈ G is denoted ord(g), and we use φ H : G → G/H to denote the natural homomorphism. We use gcd(a, b) to denote the greatest common divisor of a, b ∈ Z. 2000 Mathematics Subject Classification. 11B75 (20K01).

Abstract. 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.

Let G be an abelian group of order n and let μ be a sequence of elements of G with length 2n−k+1 taking k distinct values. Assuming that no value occurs n−k+3 times, we prove that the sums of the n-subsequences of μ must include a non-null subgroup. As a corollary we show that if G is cyclic then μ has an n-subsequence summing to 0. This last result, conjectured by Bialostocki, reduces to the Erdos–Ginzburg–Ziv theorem for k=2.

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