Master of Science, University of Akron, 2010, Mathematics

William Burnside's paqb theorem is a very important result in group theory, which states that any group G of order paqb is solvable. An interesting fact about this theorem is that it was originally proven using techniques from character theory, another branch of algebra. In fact, it was about seventy years before a group-theoretic proof of Burnside's theorem was developed through the work of Goldschmidt, Matsuyama,Bender, and other mathematicians. Their approach to proving the theorem was to show that, in essence, minimally simple groups of size paqb do not exist. Our purpose here is to use various techniques from the group-theoretic proof of Burnside's theorem to establish and prove similar results about minimally simple groups G of arbitrary order.

Committee: James Cossey Dr. (Advisor); Jeffrey Riedl PhD (Committee Member); Antonio Quesada PhD (Committee Member) Subjects: Mathematics

PHD, Kent State University, 2014, College of Arts and Sciences / Department of Mathematical Sciences

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The character degree graph Δ(G) of G is the graph whose set of vertices is the set of primes dividing an element of cd(G), with an edge between p and q if pq divides some element of cd(G). We say G is almost simple provided S ≤ G ≤ Aut(S) for some simple group S, which is the socle of G, Soc(G). Here we determine the character degree graphs of almost simple groups whose socle is one of PSL3(q), PSU3(q2), or Sz(q). We also discuss Δ(G) for almost simple G with other socles.

Committee: Donald White (Advisor); Stephen Gagola (Committee Member); Mark Lewis (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

Doctor of Philosophy, The Ohio State University, 2012, Mathematics

We study saturated fusion systems F on a finite 2-group S with an involution centralizer having a unique component on a dihedral group and containing the Baumann subgroup of S. Assuming F is perfect with no nontrivial normal 2-subgroups and the centralizer of the component is a cyclic 2-group, it is shown F is uniquely determined as the 2-fusion system of L_4(q) for some q = 3 (mod 4). This should be viewed as a contribution to a program recently outlined by Aschbacher for the classification of simple fusion systems at the prime 2. The analogous problem in the classification of finite simple groups of component type (the L_2(q), A_7 standard component problem) was one of the last to be completed, and was ultimately only resolved in an inductive context with heavy machinery. Thanks primarily to the hypothesis concerning the Baumann subgroup and the absence of cores, our arguments by contrast require only 2-fusion analysis and transfer. We prove a generalization of the Thompson transfer lemma in the context of fusion systems, which is applied often.

Committee: Ronald Solomon (Advisor); Matthew Kahle (Committee Member); Jean-Francois Lafont (Committee Member); Richard Lyons (Committee Member) Subjects: Mathematics

PHD, Kent State University, 2008, College of Arts and Sciences / Department of Mathematical Sciences

In the late 1990s, Bertram Huppert posed a conjecture which, if true, would sharpen the connection between the structure of a nonabelian finite simple group H and the set of its characterdegrees. Specifically, Huppert made the following conjecture. Huppert's Conjecture: Let G be a finite group and H a finite nonabelian simple group such that the sets of character degrees of G and H are the same. Then G is isomorphic to the direct product of H and A, where A is an abelian group. To lend credibility to his conjecture, Huppert verified it on a case-by-case basis for many nonabelian simple groups, including the Suzuki groups, many of the sporadic simple groups, and a few of the simple groups of Lie type. Except for the Suzuki groups and the family of simple groups PSL2(q), for q>3 prime or a power of a prime, Huppert proves the conjecture for specific simple groups of Lie type of small, fixed rank. We extend Huppert's results to all the linear, unitary, symplectic, and twisted Ree simple groups of Lie type of rank two. In this dissertation, we will verify Huppert's Conjecture for 2G2(q2) for q2=32m+1, m>0, and show progress toward the verification of Huppert's Conjecture for the simple group G2(q) for q>4. We will also outline our extension of Huppert's results for the remaining families of simple groups of Lie type of rank two, namely PSL3(q) for q>8, PSU3(q2) for q>9, and PSp4(q) for q>7.

Committee: Donald White (Advisor); Stephen Gagola Jr. (Committee Member); Mark Lewis (Committee Member); Michael Mikusa (Committee Member); Arden Ruttan (Committee Member) Subjects: Mathematics