The Spread of Almost Simple Classical Groups

This monograph studies generating sets of almost simple classical groups by bounding their spread. It proves a conjecture of Breuer, Guralnick, and Kantor for these groups, stating that they are 3/2-generated if and only if every proper quotient is cyclic. The first half of the monograph focuses on analyzing the automorphisms, fixed point ratios, and subgroup structure of almost simple classical groups, while the second half discusses Shintani descent. The monograph is of interest to researchers in group generation and provides a general introduction to almost simple classical groups.

Format: Paperback / softback
Length: 154 pages
Publication date: 26 May 2021
Publisher: Springer Nature Switzerland AG

This monograph delves into the study of generating sets of almost simple classical groups, focusing on limiting the spread of these groups. Guralnick and Kantor addressed a 1962 question posed by Steinberg by demonstrating that in a finite simple group, every nontrivial element is part of a generating pair. These groups are referred to as 3/2-generated. Breuer, Guralnick, and Kantor proposed a conjecture that a finite group is 3/2-generated if and only if every proper quotient is cyclic. In this monograph, we prove a strong version of this conjecture for almost simple classical groups by bounding the spread of these groups. This involves analyzing the automorphisms, fixed point ratios, and subgroup structure of almost simple classical groups, with the first half of the monograph dedicated to these general topics. Specifically, we provide a comprehensive exposition of Shintani descent. While this monograph primarily appeals to researchers interested in group generation, the introductory chapters also serve as a comprehensive introduction to the almost simple classical groups.

Weight: 454g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030740993
Edition number: 1st ed. 2021