Doctor of Philosophy, The Ohio State University, 2008, Mathematics

This dissertation studies the ‘asymptotic existence' conjecture for 3-designs with the primary goal of constructing new families of 3-designs. More specifically, this dissertation includes the following:Firstly, by considering the action of the group PSL(2,q) on the finite projective line and the orbits of the action of this group to construct simple 3-designs. While the case q congruent to 3 modulo 4 is 3-homogeneous (so that orbits of any ‘base' block' would yield designs), the case q congruent to 1 modulo 4 does not work the same way. We overcome some of these issues by considering appropriate unions of orbits to produce new infinite families of 3-designs with PSL(2,q) acting as a group of automorphisms. We also prove that our constructions actually produce an abundance of simple 3-designs for any block size if q is sufficiently large and also construct a large set of Divisible designs as an application of our constructions. We generalize the notion of a Candelabra system to more general structures, called Rooted Forest Set systems and prove a few general results on combinatorial constructions for these general set structures. Then, we specialize to the case of k=6 and extend a theorem of Hanani to produce new infinite families of Steiner 3-designs with block size 6. Finally, we consider Candelabra systems and prove that a related incidence matrix has full row rank over the rationals. This leads to interesting possibilities for ‘lambda large' theorems for Candelabra systems. While a ‘lambda large' theorem for Candelabra systems do not directly yield any Steiner 3-design, it allows for constructions of new Steiner 3-designs on large sets using methods such as Block spreading.

Committee: Neil Robertson PhD (Advisor); Akos Seress PhD (Committee Member); Stephen Milne PhD (Committee Member) Subjects: Mathematics

Master of Science, University of Akron, 2013, Mathematics

Let G be a finite group that is equal to its commutator subgroup, and suppose that G contains an element of order 2 whose centralizer in G is dihedral of 2-power order. We study the cases where this centralizer is dihedral of order 8, 16, 32, 64, 128, or 256. It is true in each case that this centralizer is a Sylow 2-subgroup of G. We then use character-theoretic techniques to generate a list of possibilities for the order of G. In the process of generating this list of possible orders, we prove several results about the structure of our group under consideration. We then strengthen the original hypotheses to require G to be non-abelian simple, and we use the results proved about the structure of G to eliminate all possible orders such that there is no non-abelian simple group of that order.

Committee: Jeffrey Riedl Dr. (Advisor); James Cossey Dr. (Committee Member); Hung Nguyen Dr. (Committee Member) Subjects: Mathematics