The Spread of Almost Simple Classical Groups

Beschreibung

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. 

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. 

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

Autor*in

Scott Harper

Themen in »The Spread of Almost Simple Classical Groups«

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

Stimmen zu »The Spread of Almost Simple Classical Groups«

“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)

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