4 research outputs found
Some new results about a conjecture by Brian Alspach
In this paper we consider the following conjecture, proposed by Brian
Alspach, concerning partial sums in finite cyclic groups: given a subset of
of size such that ,
it is possible to find an ordering of the elements of
such that the partial sums , , are nonzero
and pairwise distinct. This conjecture is known to be true for subsets of size
in cyclic groups of prime order. Here, we extend such result to any
torsion-free abelian group and, as a consequence, we provide an asymptotic
result in .
We also consider a related conjecture, originally proposed by Ronald Graham:
given a subset of , where is a prime, there
exists an ordering of the elements of such that the partial sums are all
distinct. Working with the methods developed by Hicks, Ollis and Schmitt, based
on the Alon's combinatorial Nullstellensatz, we prove the validity of such
conjecture for subsets of size
Cycle type in Hall-Paige: A proof of the Friedlander-Gordon-Tannenbaum conjecture
An orthomorphism of a finite group is a bijection
such that is also a bijection. In 1981, Friedlander,
Gordon, and Tannenbaum conjectured that when is abelian, for any
dividing , there exists an orthomorphism of fixing the identity and
permuting the remaining elements as products of disjoint -cycles. We prove
this conjecture for all sufficiently large groups.Comment: 34 page