This book gives a proof of Cherlin’s conjecture for finite binary primitive permutation groups. Motivated by the part of model theory concerned with Lachlan’s theory of finite homogeneous relational structures, this conjecture proposes a classification of those finite primitive permutation groups that have relational complexity equal to 2.
The first part gives a full introduction to Cherlin’s conjecture, including all the key ideas that have been used in the literature to prove some of its special cases. The second part completes the proof by dealing with primitive permutation groups that are almost simple with socle a group of Lie type. A great deal of material concerning properties of primitive permutation groups and almost simple groups is included, and new ideas are introduced.
Addressing a hot topic which cuts across the disciplines of group theory, model theory and logic, this book will be of interest to a wide range of readers. It will be particularly useful for graduate students and researchers who need to work with simple groups of Lie type.
Author(s): Nick Gill, Martin W. Liebeck, Pablo Spiga
Series: Lecture Notes in Mathematics, 2302
Publisher: Springer
Year: 2022
Language: English
Pages: 220
City: Cham
Contents
1 Introduction
1.1 Basics: The Definition of Relational Complexity
1.1.1 Relational Structures
1.1.2 Tuples
1.2 Basics: Some Key Examples
1.2.1 Existing Results on Relational Complexity
1.3 Motivation: On Homogeneity
1.3.1 Existing Results on Homogeneity
1.4 Motivation: On Model Theory
1.5 Motivation: Other Important Statistics
1.6 Beyond the Binary Conjecture
1.7 Methods: Basic Lemmas
1.7.1 Relational Complexity and Subgroups
1.7.2 Relational Complexity and Subsets
1.7.3 Strongly Non-binary Subsets
1.8 Methods: Frobenius Groups
1.9 Methods: On Computation
2 Preliminary Results for Groups of Lie Type
2.1 Results on Alternating Sections
2.2 Stabilizers Containing Certain Elements
2.2.1 Some Groups that Are Not Almost Simple
2.2.2 Classical Groups
2.2.3 Exceptional Groups
2.3 Results on Odd-Degree Actions
2.4 Results on Centralizers
2.5 Outer Automorphisms of Groups of Lie Type
2.6 On Fusion and Factorization
3 Exceptional Groups
3.1 Maximal Subgroups of Exceptional Groups of Lie Type
3.2 Small Exceptional Groups of Lie Type
3.3 Maximal Subgroups in (I) of Theorem 3.1.1: Parabolic Subgroups
3.4 Type (II), Part 1: Reductive Subgroups of Maximal Rank
3.5 Type (II), Part 2: Maximal Torus Normalizers
3.6 Maximal Subgroups in (V) of Theorem 3.1.1
3.7 Maximal Subgroups in (VI) of Theorem 3.1.1
3.8 The Remaining Families in Theorem 3.1.1
3.8.1 Type (III)
3.8.2 Type (IV)
3.8.3 Type (VII)
3.8.4 Type (VIII)
3.8.5 Type (IX)
3.8.6 Type (X)
4 Classical Groups
4.1 Background on Classical Groups
4.1.1 Basic Assumptions
4.2 Family C2
4.2.1 Case S=SL(n,q)
4.2.2 The Totally Singular Case
4.2.3 A General Reduction
4.2.4 Case Where S=SUn(q) and the Wi Are Non-degenerate
4.2.5 Case Where S=Spn(q) and the Wi Are Non-degenerate
4.2.6 Case Where S=Omega(n,q) for nq Odd, and the Wi Are Non-degenerate
4.2.7 Case Where S=Ωn(q) and the Wi Are Non-degenerate
4.3 Family C3
4.3.1 Case S=SL(n,q)
4.3.2 Case S=SU(n,q)
4.3.3 Case S=Sp(n,q)
4.3.4 Case S=Omega(n,q)
4.4 Family C4
4.4.1 Case S=SL(n,q)
4.4.2 Case S=SU(n,q)
4.4.3 Case Where S Is Symplectic or Orthogonal
4.4.4 The Remaining Cases
4.5 Family C5
4.5.1 Case S=SL(n,q)
4.5.2 Case S=SU(n,q)
4.5.3 Case S=Sp(n,q)
4.5.4 Case S Is Orthogonal
4.6 Family C6
4.7 Family C7
4.7.1 Case S=SL(n,q)
4.7.2 Case S=SU(n,q)
4.7.3 Case S=Sp(n,q)
4.7.4 Case S=Omega(n,q)
4.7.5 Case S=Omega+(n,q)
4.8 Family C8
4.8.1 Case S=SL(n,q)
4.8.2 Case S=Sp(n,q)
4.9 Family S
4.9.1 Strategies
Strategy 1: Subgroups Containing Centralizers
Strategy 2: Odd Degree Actions
Strategy 3: Using Distinguished Elements
4.9.2 The Case Where M0 Is Alternating
4.9.3 The Case Where M0 Is Sporadic
4.9.4 The Case Where M0 Is of Lie Type
4.10 Exceptional Automorphisms
Bibliography
