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Subsets of finite groups exhibiting additive regularity
In this article we aim to develop from first principles a theory of sum sets
and partial sum sets, which are defined analogously to difference sets and
partial difference sets. We obtain non-existence results and characterisations.
In particular, we show that any sum set must exhibit higher-order regularity
and that an abelian sum set is necessarily a reversible difference set. We next
develop several general construction techniques under the hypothesis that the
over-riding group contains a normal subgroup of order 2. Finally, by exploiting
properties of dihedral groups and Frobenius groups, several infinite classes of
sum sets and partial sum sets are introduced
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
Some results on partial difference sets and partial geometries
This thesis shows results on 3 different problems involving partial difference sets (PDS) in abelian groups, and uses PDS to study partial geometries with an abelian Singer group. First, the last two undetermined cases of PDS on abelian groups with k ≤ 100, both of order 216, were shown not to exist. Second, new parameter bounds for k and ∆ were found for PDS on abelian groups of order p^n , p an odd prime, n odd. A parameter search on p^5 in particular was conducted, and only 5 possible such cases remain for p \u3c 250. Lastly, the existence of rigid type partial geometries with an abelian Singer group are examined; existence is left undetermined for 11 cases with α ≤ 200. This final study led to the determination of nonexistence for an infinite class of cases which impose a negative Latin type PDS
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