Root involutions

Abstract

In this chapter we will describe how the classification of finite groups generated by a class D of root involutions can be reduced to the classification of groups generated by abstract root subgroups. This root involution classification played an important role in the classification of finite simple groups and also for the determination of the subgroup structure of finite simple groups. Apart from its use for identifications of simple groups of characteristic 2 type, the main application of the root involution classification is through the theory of TI-subgroups, i.e. elementary abelian 2-subgroups T satisfying T ∩ T ^g = T or 1 for all g ∈G, see exercises III(3.23)(4)–(6). For a more detailed discussion of the role of TI-subgroups for the classification of finite simple groups see Aschbachers article [Asc80]. Also the special case of {3,4}⁺-transpositions is of importance for the classification of groups with large extraspecial 2-subgroup, see [Tim78a] and [Smi80a], which was considered as one of the main open problems for the classification of finite simple groups, since most of the sporadic groups satisfy this condition.