## Zero-sum problems in finite cyclic groups

##### Abstract

The purpose of this thesis is to investigate some open problems in the area of
combinatorial number theory referred to as zero-sum theory. A zero-sequence in a
finite cyclic group G is said to have the basic property if it is equivalent under group
automorphism to one which has sum precisely IGI when this sum is viewed as an
integer. This thesis investigates two major problems, the first of which is referred
to as the basic pair problem. This problem seeks to determine conditions for which
every zero-sequence of a given length in a finite abelian group has the basic property.
We resolve an open problem regarding basic pairs in cyclic groups by demonstrating
that every sequence of length four in Zp has the basic property, and we conjecture
on the complete solution of this problem. The second problem is a 1988 conjecture
of Kleitman and Lemke, part of which claims that every sequence of length n in Zn
has a subsequence with the basic property. If one considers the special case where n
is an odd integer we believe this conjecture to hold true. We verify this is the case
for all prime integers less than 40, and all odd integers less than 26. In addition,
we resolve the Kleitman-Lemke conjecture for general n in the negative. That is, we
demonstrate a sequence in any finite abelian group isomorphic to Z2p (for p ~ 11
a prime) containing no subsequence with the basic property. These results, as well
as the results found along the way, contribute to many other problems in zero-sum
theory.