Subgroups of exceptional groups of Lie type generated by long root elements, II. Characteristic two

Abstract

Let G be an exceptional group of Lie type over a finite field F,, q = p”, and assume the Weyl rank of G is at least four. Let X be the collection of long root subgroups of G, and r the collection of long root elements of G. In [7,8 J we determined those subgroups L of G generated by long root subgroups with O,(L) = 1. This was done by studying the geometry of G acting on S’, with classification theorems only used in [7] to identify the possible isomorphism classes for L. In Part I [9] of this series we determined those subgroups L of G generated by long root elements with O,(L) = 1 where p > 2. In that paper we used Aschbacher’s powerful result classifying the groups in Chev(p) for p odd (21 in order to identify L. In this paper we complete the determination in the case p = 2, using the theorems of Aschbacher [ 11, Fischer [ 131 and Timmesfeld [ 181 to identify L. The determination of the subgroups of classical groups generated by elements central in a long root subgroup was carried out by Kantor [ 151 in all characteristics using the aforementioned classification theorems. For references on earlier attempts at these determinations for the classical groups see the introduction of [15]. In another paper [ 111 we make use of our results to determine the minimal degree of a permutation representation of an exceptional group. We also foresee these results having application to the more general problem of determining the maximal subgroups of an exceptional group.