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
CONTENT
I Rank One Groups -- ยง 1 Definition, examples, basic properties -- ยง 2 On the structure of rank one groups -- ยง 3 Quadratic modules -- ยง 4 Rank one groups and buildings -- ยง 5 Structure and embeddings of special rank one groups -- II Abstract Root Subgroups -- ยง 1 Definitions and examples -- ยง 2 Basic properties of groups generated by abstract root subgroups -- ยง 3 Triangle groups -- ยง4 The radical R(G) -- ยง 5 Abstract root subgroups and Lie type groups -- III Classification Theory -- ยง 1 Abstract transvection groups -- ยง 2 The action of G on ? -- ยง 3 The linear groups and EK6 -- ยง 4 Moufang hexagons -- ยง 5 The orthogonal groups -- ยง6 D4(k) -- ยง 7 Metasymplectic spaces -- ยง8 E6(k),E7(k) and E8(k) -- ยง 9 The classification theorems -- IV Root involutions -- ยง 1 General properties of groups generated by root involutions -- ยง 2 Root subgroups -- ยง 3 The Root Structure Theorem -- ยง 4 The Rank Two Case -- V Applications -- ยง 1 Quadratic pairs -- ยง 2 Subgroups generated by root elements -- ยง3 Local BN-pairs -- References -- Symbol Index
