Overview
- Introduces the almost simple groups together with their maximal subgroups and automorphisms
- Provides a very well-written, comprehensive account of Shintani descent for applications in group theory in a useful context
- Classifies the finite 3/2-generated groups in the important and substantial case of almost simple classical groups
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2286)
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About this book
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.
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Keywords
Table of contents (6 chapters)
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Front Matter
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Back Matter
Reviews
“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)
Authors and Affiliations
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.
Bibliographic Information
Book Title: The Spread of Almost Simple Classical Groups
Authors: Scott Harper
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-030-74100-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Softcover ISBN: 978-3-030-74099-3Published: 26 May 2021
eBook ISBN: 978-3-030-74100-6Published: 25 May 2021
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: VIII, 154
Number of Illustrations: 35 b/w illustrations
Topics: Group Theory and Generalizations