Groups, Combinatorics and Geometry

The following sections are included:

• Preface

• CONTENTS

• List of authors and addresses

The following sections are included:

• Introduction

• Preliminaries

• The amalgam method
  • The issues
  • Definability
  • Application of the amalgam method (the Moufang property)

• Identification

• References

We demonstrate that there is a relation between the Curtis-Tits theorem and Phan's theorems that goes beyond the similarity in appearance. We present a geometric construction connecting those theorems and suggesting that Phan's theorems can be thought of as “twisted versions” of the Curtis-Tits theorem. The construction itself further suggests that Phan's theorems are only some of many possible such theorems. We make this explicit by presenting a new Phan-type theorem for the symplectic groups.

The following sections are included:

• INTRODUCTION

• THE SYMMETRIC GROUP
  • Formal characters
  • Induction and restriction operators
  • The affine Kac-Moody algebra
  • The crystal graph
  • The modular branching graph
  • More on characters
  • Blocks

• THE DOUBLE COVERS
  • Analogues of the Jucys-Murphy elements
  • Formal characters
  • Induction and restriction operators
  • The affine Kac-Moody algebra
  • The crystal graph
  • The modular branching graph
  • Blocks

• REFERENCES

Coherent configurations are combinatorial objects invented for the purpose of studying finite permutation groups; every permutation group which is not doubly transitive preserves a non-trivial coherent configuration. However, symmetric coherent configurations have a much longer history, having been used in statistics under the name of association schemes.

The relationship between permutation groups and association schemes is quite subtle; there are groups which preserve no non-trivial association scheme, and other groups for which there is not a unique minimal association scheme.

This paper gives a brief outline of the theory of coherent configurations and association schemes, and reports on some recent work on the connection between association schemes and permutation groups.

The following sections are included:

• Introduction

• Basic Shuffling

• History and Practical Consequences

• Other Measures of Randomness

• Some Mathematical Connections
  • Descent Theory
  • Connections with Symmetric Function Theory
  • Work of Lalley
  • Early Shuffling

• Some Generalizations
  • Hyperplane Walks
  • Some Representation Theory
  • Brown's Semigroup Walks

• Some Open Problems

• REFERENCES

We investigate the proportion of fixed point free permutations (derangements) in finite transitive permutation groups. This article is the first in a series where we prove a conjecture of Shalev that the proportion of such elements is bounded away from zero for a simple finite group. In fact, there are much stronger results. This article focuses on finite Chevalley groups of bounded rank. We also discuss derangements in algebraic groups and in more general primitive groups. These results have applications in questions about probabilistic generation of finite simple groups and maps between varieties over finite fields.

The following sections are included:

• Introduction

• Permutation groups

• Matrix groups

• Nonconstructive recognition of simple groups

• Constructive recognition of simple groups

• General matrix groups

• References

The following sections are included:

• Maximal subgroups of exceptional algebraic groups

• Maximal subgroups of finite exceptional groups

• References

The following sections are included:

• Introduction

• General bounds

• Conjectures of Babai and Cameron

• Pyber's Conjecture

• References

The following sections are included:

• Introduction

• Notation and Hypothesis

• The Modules
  • Results
  • An Example

• The Local Analysis
  • Pushing Up
  • Groups with a Proper p-Core
    • Strongly p-embedded subgroups
    • CGT
  • ¬E!
    • The open “¬E!, b = 1”-Problem
  • E!
    • The Structure Theorem
      • The Structure Theorem for Y_M ≤ Q
      • The Structure Theorem for Y_M ≰ Q
    • The P!-Theorem
    • The -Theorem
    • The Small World Theorem
      • The open “gb = 2”-Problem
    • Rank 2
      • The open “Rank 2, gb=1”-Problem
    • gb = 1
      • The open “ ,gb=1”-Problem

• The Global Analysis

• References

The following sections are included:

• INTRODUCTION

• THE DESCENT PRINCIPLE AND SOME APPLICATIONS
  • The descent principle
  • Divisibility properties determined by free normal subgroups
  • A reduction formula
  • A possible generalization of the descent principle

• A CRITERION FOR PERIODICITY MODULO p

• A FUNCTIONAL EQUATION ASSOCIATED WITH THE GROUPS
  • A functional equation for X_G,q(z)
  • Some remarks concerning assertions (A) and (B)
  • The coefficients

• THE PATTERNS II^(p)
  • The case where G is in Fin(p)
  • Rationality
  • Divisibility patterns and Fermat primes

• FREE PARITY PATTERNS IN HECKE GROUPS
  • The type τ_q and identity (54)
  • The patterns
  • An afterthought

• PARITY PATTERNS IN ONE-RELATOR GROUPS
  • The result
  • Some background: A recurrence relation modulo 2, Euler's pentagonal theorem, and results concerning the parity of the partition function
  • Some representation theory

• REFERENCES

We aim to count the number of conjugacy classes of nets, i.e. triples of 6-transpositions in the Monster up to braiding. An exact answer is given subject to a conjecture.

A permutation group is quasiprimitive if each of its non-trivial normal subgroups is transitive. Quasiprimitive permutation groups arise naturally when studying automorphism groups of vertex-transitive graphs, and they form a family of permutation groups that properly contains all primitive permutation groups. In this chapter we describe the current state of our knowledge of the structure of finite quasiprimitive groups and in particular of the quasiprimitive permutation groups containing a given finite quasiprimitive group. To illustrate how these results can be used in graph theory we discuss their application to finite s-arc-transitive graphs and other classes of vertex-transitive graphs. Many of the results depend on the finite simple group classification.

In 1937 B. H. Neumann gave a simple construction of continuously many non-isomorphic finitely generated groups, using families of finite alternating groups. We describe a generalisation of this construction, designed to settle various open problems in the area of subgroup growth.

As an unexpected byproduct our groups yield new examples related to a question of Grothendieck concerning isomorphism of groups with isomorphic profinite completions.

We also give a quick survey of some areas of infinite group theory in which these and related constructions based on finite simple groups play a role.

The following sections are included:

• INTRODUCTION

• FINITE QUOTIENTS OF D^× ARE SOLVABLE

• HERMITIAN SPACES AND UNITARY GROUPS, NOTATION AND DEFINITIONS

• GENERALIZED REFLECTIONS

• THE REDUCTION FROM dim(V) > 2 TO dim(V) = 2

• THE CASE WHEN dim(V) = 1

• REFERENCES

The following sections are included:

• Results

• Hints of proofs

• References

The following sections are included:

• Introduction

• Bound and gap results for symmetric groups and alternating groups

• Bound results for finite groups of Lie type

• Gap results for classical groups
  • Special linear groups
  • Special unitary groups
  • Symplectic groups in odd characteristic
  • Symplectic groups in even characteristic

• Minimal polynomial problem
  • char(F) = ℓ ≠ r case
    • Unipotent subcase: r = p
    • Semisimple subcase: r ≠ p
  • char(F) = ℓ = r case

• RANK 3 PERMUTATION MODULES
  • Case I : c ≠ d in F
  • Case II : c = d (in F)

• MORE APPLICATIONS

• REFERENCES

The following sections are included:

• Introduction

• Rank one groups

• On the structure of Lie-type groups

• The Steinberg-presentation

• The Curtis-Tits-presentation

• References

The following sections are included:

• Introduction

• Preliminaries

• The case

• The case s(G) = 3

• The case s(G) = 2, n = 3

• The correlation case

• The collineation case

• Some additional remarks

• References

We give a survey of computational methods and results concerning the Monster sporadic simple group.

There are now three computer constructions of the Monster which are proving effective in answering real questions about this group. The first construction over the field of two elements is the fastest for calculations, and has been used to show the group is a Hurwitz group.

The second construction over the field of three elements, uses an involution centralizer as the heart of the construction, and has proved to be the most useful as far as calculations with subgroups is concerned. P. E. Holmes has used this construction to find explicitly four new conjugacy classes of maximal subgroups, as well as to eliminate various other possibilities for maximal subgroups.

The third construction over the field of seven elements uses the same generators as the first construction, which means that elements given as words in these generators can be investigated modulo 2 and modulo 7 simultaneously. This gives enough information in most cases to determine the conjugacy class of the element.