This book provides the first representation theoretic and algorithmic approach to the theory of abstract finite simple groups. It presents self-contained proofs of classical and new group order formulas, and a new structure theorem for abstract finite simple groups. This, and the famous Brauer-Fowler theorem, provides the theoretical background for the author's algorithm which constructs all finite simple groups G having a 2-central involution z with a given centralizer CG(z)=H. The methods presented are designed for the construction of matrix representations, permutation representations and character tables of large finite groups. The author constructs all the simple satellites of the known simple groups that are not uniquely determined by a given centralizer H. Uniform existence and uniqueness proofs are given for the modern sporadic simple groups discovered by Janko, Higman and Sims, Harada, and Thompson. This latter result proves a long standing open problem in the theory. The experimental results (courtesy of M. Weller) for Chapter 12 are documented in the accompanying DVD.
Table of Contents
Introduction; 1. Prerequisites from group theory; 2. Group representations and character theory; 3. Modular representation theory; 4. Group order formulas and structure theorem; 5. Permutation representations; 6. Concrete character tables of matrix groups; 7. Methods for constructing finite simple groups; 8. Finite simple groups with proper satellites; 9. Janko group J1; 10. Higman-Sims group HS; 11. Harada group Ha; 9. Thompson group Th; Bibliography; List of symbols; Index.
Gerhard Michler is an Emeritus Professor of the Institute of Experimental Mathematics at the Univerity of Duisburg-Essen