(2,3)-Generation of Exceptional Groups

Abstract

We study two aspects of generation of large exceptional groups of Lie type. First we show that any finite exceptional group of Lie rank at least four is (2,3)-generated, that is, a factor group of the modular group PSL2(Z). This completes the study of (2,3)-generation of groups of Lie type. Second, we complete the proof that groups of type E7 and E8 over fields of odd characteristic occur as Galois groups of geometric extensions of Qab(t), where Qab denotes the maximal Abelian extension field of Q. Finally, we show that all finite simple exceptional groups of Lie type have a pair of strongly orthogonal classes. The methods of proof in all three cases are very similar and require the Lusztig theory of characters of reductive groups over finite fields as well as the classification of finite simple groups.