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.