Abstract
Let G ⩽ Sym(Ω) be a permutation group on a finite set Ω. A base for G is a subset of Ω withtrivial pointwise stabiliser, and the minimal size of a base is called the base size of G, denoted
b(G). This classical invariant has found a wide range of applications and connections, attracting
significant attention since the early years of group theory in the 19th century. Historically, there
has been a particular focus on studying base sizes for primitive groups, which can be viewed as
the basic building blocks of all finite permutation groups, and this still remains a very active area
of research.
In the 1990s, Jan Saxl initiated a project with the ultimate goal of classifying the primitive
groups G with b(G) = 2. In order to study the bases for these groups, Burness and Giudici defined
the Saxl graph of G, where the vertex set is Ω, and two vertices are adjacent if they form a base.
This opened up a new direction of studying the graph-theoretical properties of the Saxl graphs of
primitive groups.
In this thesis, we focus on the study of base sizes and Saxl graphs of primitive groups. We
determine the exact base size of every primitive group of diagonal type, and this is the first
family of primitive groups arising in the O’Nan-Scott theorem for which the precise base size is
known. We also initiate the study of base sizes for product type primitive groups, focussing on the
groups with soluble point stabilisers. In addition, we extend the definition of the Saxl graph to
groups G with b(G) ⩾ 3 and we study various connectivity properties of this graph for primitive
groups, including its diameter, arc-transitivity and completeness. We adopt probabilistic and
computational methods in order to establish the main theorems, which rely on a detailed analysis
of the conjugacy classes and subgroup structure of the finite almost simple groups.
| Date of Award | 13 May 2025 |
|---|---|
| Original language | English |
| Awarding Institution |
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| Supervisor | Tim Burness (Supervisor) |
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