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Character Theoretic Techniques for Nonabelian partial Difference Sets
Nelson, Seth R
Nelson, Seth R
Abstract
A(v,k,λ,µ)-partial difference set (PDS) is a subset D of size k of a group G of order v such that every nonidentity element g of G can be expressed in either λ or µ different ways as a product xy−1, x,y ∈ D, depending on whether or not g is in D. If D is inverse closed and 1 / ∈ D, then the Cayley graph Cay(G,D) is a (v,k,λ,µ)-strongly regular graph (SRG). PDSs have been studied extensively over the years, especially in abelian groups, where techniques from character theory have proven to be particularly effective. Recently, there has been considerable interest in studying PDSs in nonabelian groups, and the purpose of this thesis is to develop character theoretic techniques that apply in the nonabelian setting. We prove that analogues of character theoretic results of Ott [25] about generalized quadrangles of order s also hold in the general PDS setting. In conjunction with Ott’s results, we further develop character theoretic techniques to compute the intersection of a PDS with the conjugacy classes of the parent group. With these techniques, we are able to prove the nonexistence of PDSs in numerous instances. Furthermore, we are able to use these techniques constructively, computing several examples of PDSs in nonabelian groups not previously recognized in the literature, including an infinite family of genuinely nonabelian PDSs associated to the block-regular Steiner 2-designs first studied by Wilson
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2025-04-01
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Mathematics