9,846 research outputs found
On the Olson and the Strong Davenport constants
A subset of a finite abelian group, written additively, is called
zero-sumfree if the sum of the elements of each non-empty subset of 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, -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 -groups of rank at most , paralleling and building
on recent results on this problem for the Olson constant
On the index of length four minimal zero-sum sequences
Let be a finite cyclic group. Every sequence over can be written
in the form where and n_1, \ldots,
n_l\in[1, \ord(g)], and the index \ind(S) of is defined to be the
minimum of (n_1+\cdots+n_l)/\ord(g) over all possible such that
.
A conjecture on the index of length four sequences says that every minimal
zero-sum sequence of length 4 over a finite cyclic group with has index 1. The conjecture was confirmed recently for the case when
is a product of at most two prime powers. However, the general case is
still open. In this paper, we make some progress towards solving the general
case. Based on earlier work on this problem, we show that if is a finite cyclic group of order such that and
is a minimal zero-sum sequence over such that
with , and
for some , then \ind(S)=1.
By using an innovative method developed in this paper, we are able to give a
new (and much shorter) proof to the index conjecture for the case when is
a product of two prime powers
The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup
Let be a finite group written multiplicatively. By a sequence over ,
we mean a finite sequence of terms from 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 . The small
Davenport constant is the maximal integer such that
there is a sequence over of length which has no nontrivial,
product-one subsequence. The large Davenport constant 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 , and
if is abelian, then equality holds. However, for non-abelian groups, these
constants can differ significantly. Now suppose has a cyclic, index 2
subgroup. Then an old result of Olson and White (dating back to 1977) implies
that if is non-cyclic, and
if is cyclic. In this paper, we determine the large Davenport constant of
such groups, showing that , where is the commutator subgroup of
Scenery Reconstruction on Finite Abelian Groups
We consider the question of when a random walk on a finite abelian group with
a given step distribution can be used to reconstruct a binary labeling of the
elements of the group, up to a shift. Matzinger and Lember (2006) give a
sufficient condition for reconstructibility on cycles. While, as we show, this
condition is not in general necessary, our main result is that it is necessary
when the length of the cycle is prime and larger than 5, and the step
distribution has only rational probabilities. We extend this result to other
abelian groups.Comment: 16 pages, 2 figure
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