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Partial Difference Sets in Nonabelian Groups and Strongly Regular Cayley Graphs
Tauscheck, Gabrielle
Tauscheck, Gabrielle
Abstract
A regular graph $\Gamma$ with $v$ vertices and valency $k$ is said to be a $(v,k,\lambda, \mu)$-strongly regular graph if any two adjacent vertices are both joined to exactly $\lambda$ other vertices and two nonadjacent vertices are both joined to exactly $\mu$ other vertices. Let $G$ be a group of order $v$ and $D$ a $k$-element subset of $G$. Then $D$ is called a $(v,k,\lambda,\mu)$-partial difference set if for every nonidentity element $g$ of $D$, the equation $d_1d_2^{-1}=g$ has exactly $\lambda$ solutions $(d_1,d_2) \in D \times D$; and for every nonidentity element $g'$ of $G$ not in $D$, the equation $d_1d_2^{-1}=g'$ has exactly $\mu$ solutions. It is known that a subset $D$ of $G$ with $e \notin D$ and $\{d^{-1} | d \in D\}=D$ is a partial difference set if and only if the Cayley graph generated by $D$ is strongly regular. Yoshiara has given two lemmas that describe the conditions needed for an automorphism group to act regularly on a finite generalized quadrangle. De Winter, Kamischke, and Wang build upon the work of Benson to construct partial difference sets in abelian groups. In this work, we confirm Yoshiara's results, and use De Winter, Kamischke, and Wang's result in place of Benson's to generalize Yoshiara's results to nonabelian groups. In the process, we are able to rule out the existence of many partial difference sets in nonabelian groups.
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Date
2019-04-01
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Mathematics