Overview

"It was already in 1964 [Fis66] when B. Fischer raised the question: Which finite groups can be generated by a conjugacy class D of involutions, the product of any two of which has order 1, 2 or 37 Such a class D he called a class of 3-tmnspositions of G. This question is quite natural, since the class of transpositions of a symmetric group possesses this property. Namely the order of the product (ij)(kl) is 1, 2 or 3 according as {i,j} n {k,l} consists of 2,0 or 1 element. In fact, if I{i,j} n {k,I}1 = 1 and j = k, then (ij)(kl) is the 3-cycle (ijl). After the preliminary papers [Fis66] and [Fis64] he succeeded in [Fis71J, [Fis69] to classify all finite ""nearly"" simple groups generated by such a class of 3-transpositions, thereby discovering three new finite simple groups called M(22), M(23) and M(24). But even more important than his classification theorem was the fact that he originated a new method in the study of finite groups, which is called ""internal geometric analysis"" by D. Gorenstein in his book: Finite Simple Groups, an Introduction to their Classification. In fact D. Gorenstein writes that this method can be regarded as second in importance for the classification of finite simple groups only to the local group-theoretic analysis created by J. Thompson."

Full Product Details

Author:   Franz G. Timmesfeld
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2001 ed.
Volume:   95
Dimensions:   Width: 17.00cm , Height: 2.10cm , Length: 24.40cm
Weight:   0.703kg
ISBN:  

9783034875967

ISBN 10:   3034875967
Pages:   389
Publication Date:   12 June 2012
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

I Rank One Groups.- § 1 Definition, examples, basic properties.- § 2 On the structure of rank one groups.- § 3 Quadratic modules.- § 4 Rank one groups and buildings.- § 5 Structure and embeddings of special rank one groups.- II Abstract Root Subgroups.- § 1 Definitions and examples.- § 2 Basic properties of groups generated by abstract root subgroups.- § 3 Triangle groups.- §4 The radical R(G).- § 5 Abstract root subgroups and Lie type groups.- III Classification Theory.- § 1 Abstract transvection groups.- § 2 The action of G on ?.- § 3 The linear groups and EK6.- § 4 Moufang hexagons.- § 5 The orthogonal groups.- §6 D4(k).- § 7 Metasymplectic spaces.- §8 E6(k),E7(k) and E8(k).- § 9 The classification theorems.- IV Root involutions.- § 1 General properties of groups generated by root involutions.- § 2 Root subgroups.- § 3 The Root Structure Theorem.- § 4 The Rank Two Case.- V Applications.- § 1 Quadratic pairs.- § 2 Subgroups generated by root elements.- §3 Local BN-pairs.- References.- Symbol Index.

Reviews

The book is well written: the style is concise but not hard and most of the book is not too difficult to read for a graduate student. Some parts of it are certainly suited for a class. --Mathematical Reviews

The book is well written: the style is concise but not hard and most of the book is not too difficult to read for a graduate student. Some parts of it are certainly suited for a class. --Mathematical Reviews

The book is well written: the style is concise but not hard and most of the book is not too difficult to read for a graduate student. Some parts of it are certainly suited for a class. --Mathematical Reviews

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