140 research outputs found
Genuinely nonabelian partial difference sets
Strongly regular graphs (SRGs) provide a fertile area of exploration in
algebraic combinatorics, integrating techniques in graph theory, linear
algebra, group theory, finite fields, finite geometry, and number theory. Of
particular interest are those SRGs with a large automorphism group. If an
automorphism group acts regularly (sharply transitively) on the vertices of the
graph, then we may identify the graph with a subset of the group, a partial
difference set (PDS), which allows us to apply techniques from group theory to
examine the graph. Much of the work over the past four decades has concentrated
on abelian PDSs using the powerful techniques of character theory. However,
little work has been done on nonabelian PDSs. In this paper we point out the
existence of \textit{genuinely nonabelian} PDSs, i.e., PDSs for parameter sets
where a nonabelian group is the only possible regular automorphism group. We
include methods for demonstrating that abelian PDSs are not possible for a
particular set of parameters or for a particular SRG. Four infinite families of
genuinely nonabelian PDSs are described, two of which -- one arising from
triangular graphs and one arising from Krein covers of complete graphs
constructed by Godsil \cite{Godsil_1992} -- are new. We also include a new
nonabelian PDS found by computer search and present some possible future
directions of research.Comment: 24 page
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