The Spread of Almost Simple Classical Groups (Lecture Notes in Mathematics) - Softcover

Harper, Scott

Softcover

ISBN 10: 3030740994 ISBN 13: 9783030740993

Publisher: Springer, 2021

Synopsis

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. 

About this title

About the Author

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. 

From the Back Cover

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 the opening chapters also serve as a general introduction to the almost simple classical groups. 

Edition details

• Publisher: Springer
• Publication date: 2021
• Language: English
• ISBN 10: 3030740994
• ISBN 13: 9783030740993
• Binding: Paperback
• Edition number: 1
• Number of pages: 164