The Spread of Almost Simple Classical Groups

1. Front Matter

   Pages i-viii

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2. Introduction

   • Scott Harper

   Pages 1-7

3. Preliminaries

   • Scott Harper

   Pages 9-21

4. Shintani Descent

   • Scott Harper

   Pages 23-32

5. Fixed Point Ratios

   • Scott Harper

   Pages 33-52

6. Orthogonal Groups

   • Scott Harper

   Pages 53-110

7. Unitary Groups

   • Scott Harper

   Pages 111-141

8. Back Matter

   Pages 143-154

   PDF

This monograph studies generating sets of almost simple classical groups, by bounding the spread of these groups.  
 
Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group, every nontrivial element belongs to a generating pair. Groups with this property are said to be 3/2-generated. Breuer, Guralnick and Kantor conjectured that a finite group is 3/2-generated if and only if every proper quotient is cyclic. We prove a strong version of this conjecture for almost simple classical groups, by bounding the spread of these groups. This involves analysing the automorphisms, fixed point ratios and subgroup structure of almost simple classical groups, so the first half of this monograph is dedicated to these general topics. In particular, we give a general exposition of Shintani descent.  
 
This monograph will interest researchers in group generation, but theopening chapters also serve as a general introduction to the almost simple classical groups. 

• 3/2-Generation
• Almost Simple Groups
• Classical Groups
• Fixed Point Ratios
• Generating Graph
• Maximal Overgroups
• Shintani Descent
• Uniform Spread

“The monograph is essentially dedicated to experts in the field and provides a detailed analysis of the spread of the groups, as part of a wider current research project.” (Enrico Jabara, zbMATH 1510.20001, 2023)

“The purposes of the monograph are twofold. Firstly, it provides an introduction to the almost simple classical groups for a broad audience, including graduate students new to the area. Secondly, for the experts in the field, it provides a detailed analysis of the spread of these groups, as part of a wider current research project.” (Stefan Kohl, Mathematical Reviews, May, 2022)

• School of Mathematics, University of Bristol, Bristol, UK

  Scott Harper

Scott Harper is a Heilbronn Research Fellow at the University of Bristol. His main research interest is group theory, having published several papers on the subject. This is his first book. He is particularly interested in simple groups and connections between group theory and combinatorics. Previously, he was a London Mathematical Society Early Career Research Fellow at the University of Padua, and he completed his PhD at the University of Bristol, during which time he was awarded the Cecil King Travel Scholarship from the London Mathematical Society. 

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