Title:

On the lattice automorphisms of certain algebraic groups

In the first chapter we give an introduction, and a survey of known results, which we shall use throughout the dissertation. In the second chapter we first prove that every projectivity of a connected reductive nonabelian algebraic group G over K = Fp is strictly indexpreserving (Theorem 2.1.6.). Then we prove that every autoprojectivity of G induces an automorphism of the building canonically associated to O. Furthermore we show how certain autoprojectivities of G act on the Weyl group of G and on the Dynkin diagram of G. In the third chapter we restrict our attention to simple algebraic groups over K. We prove that if G is a simple algebraic group over K of rank at least 2, then the problem whether every autoprojectivity of G is induced by an automorphism, is reduced to the problem whether every autoprojectivity of G fixing every parabolic subgroup of G is the identity. Namely, if we let Γ(G) – {φε Aut L(G) I Pφ = P for every parabolic subgroup P of G} , we have Aut L(G) = Γ (Aut G)*, where (Aut G)* is the group of all autoprojectivities of G induced by an automorphism (Theorem 3.4.9. and Corollary 3.4.15.). In Chapter 4 we prove that actually Γ = {1} if G has rank at least 3 and p ≠ 2 (Theorem 4.6.5.), while in Chapter 5 we prove the same result , with different arguments, for the case of rank 1 (Corollary 5.2.6.) and 2, type A2 excluded (Corollary 5.3.8.) (for groups of rank 1 we impose no restrictions on p). Finally, in Chapter 6 we show that for the groups of type A2 Theorem 4.6.5. does not hold. For this purpose we construct a nontrivial subgroup of the group Γ(SL3(F23)) (Corollary 6.4.15.).
