17 research outputs found
Inverse zero-sum problems II
Let be an additive finite abelian group. A sequence over is called a
minimal zero-sum sequence if the sum of its terms is zero and no proper
subsequence has this property. Davenport's constant of is the maximum of
the lengths of the minimal zero-sum sequences over . Its value is well-known
for groups of rank two. We investigate the structure of minimal zero-sum
sequences of maximal length for groups of rank two. Assuming a well-supported
conjecture on this problem for groups of the form , we
determine the structure of these sequences for groups of rank two. Combining
our result and partial results on this conjecture, yields unconditional results
for certain groups of rank two.Comment: new version contains results related to Davenport's constant only;
other results will be described separatel
On product-one sequences over dihedral groups
Let be a finite group. A sequence over means a finite sequence of
terms from , where repetition is allowed and the order is disregarded. A
product-one sequence is a sequence whose elements can be ordered such that
their product equals the identity element of the group. The set of all
product-one sequences over (with concatenation of sequences as the
operation) is a finitely generated C-monoid. Product-one sequences over
dihedral groups have a variety of extremal properties. This article provides a
detailed investigation, with methods from arithmetic combinatorics, of the
arithmetic of the monoid of product-one sequences over dihedral groups.Comment: to appear in Journal of Algebra and its Application
Zero-sum problems with congruence conditions
For a finite abelian group and a positive integer , let denote the smallest integer such that
every sequence over of length has a nonempty zero-sum
subsequence of length . We determine for all when has rank at most two and, under mild
conditions on , also obtain precise values in the case of -groups. In the
same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv
constant provided that, for the -subgroups of , the Davenport
constant is bounded above by . This
generalizes former results for groups of rank two