The main purpose of this paper is to better understand groups that do not have the unique product property. In particular, the goal is to better understand Promislow's example, G, of such a group. In doing so, we will develop methods for generating examples of other sets that do not have the unique product property. With these methods we can show that there exists other distinct 14 element, square, non-unique product sets in G that are not inversions or translations. Also, this paper answers the question as to whether every non-unique product set can have only 14 elements in the negative by producing a 17 element square n.u.p. set. The secondary purpose of this paper is to demonstrate that in the group ring K[G], there are no units of support size 3. / Master of Science

In this paper, we present techniques for proving a group to be non-left-orderable or a unique product group. These methods involve the existence of a mapping from the group to R which obeys a left-multiplication criterion. By determining the existence or non-existence of such a mapping, the desired information about the group can be concluded. As examples, we apply this technique to groups of transformations in hyperbolic 2- and 3- space, and Fibonacci groups. / Master of Science

This thesis discusses the Kaplansky zero divisor conjecture. The conjecture states that a group ring of a torsion-free group over a field has no nonzero zero divisors. There are situations for which this conjecture is known to hold, such as linearly orderable groups, unique product groups, solvable groups, and elementary amenable groups. This paper considers the possibility that the conjecture is false and there is some counterexample in existence. The approach to searching for such a counterexample discussed here is to first find a torsion-free group that has subsets A and B such that AB has no unique product. We do this by exhaustively searching for the subsets A and B with fixed small sizes. When |A| = 1 or 2 and |B| is arbitrary we know that AB contains a unique product, but when |A| is larger, not much was previously known. After an example is found we then verify that the sets are contained in a torsion-free group and further investigate whether the group ring yields a nonzero zero divisor. Together with Dr. Pace P. Nielsen, assistant math professor of Brigham Young University, we created code that was implemented in Magma, a computational algebra system, for the purpose of considering each size of A and B and running through each case. Along the way we check for the possibility of torsion elements and for other conditions that lead to contradictions, such as a decrease in the size of A or B. Our results are the following: If A and B are sets of the sizes below contained in a torsion-free group, then they must contain a unique product. |A| = 3 and |B| ≤ 16; |A| = 4 and |B| ≤ 12; |A| = 5 and |B| ≤ 9; |A| = 6 and |B| ≤ 7. We have continued to run cases of larger size and hope to increase the size of B for each size of A. Additionally, we found a torsion-free group containing sets A and B, both of size 8, where AB has no unique product. Though this group does not yield a counterexample for the Kaplansky zero divisor conjecture, it is the smallest explicit example of a non-uniqueproduct group in terms of the size of A and B.