Preface

Summary

These notes grew out of a summer school on “Finite Groups and Related Geometrical Structures” held in Venice from September 5th to September 15th 2007. The aim of the course was to introduce an audience consisting mainly of PhD students and postdoctoral researchers working in finite group theory and neighboring areas to results on the subgroup structure of linear algebraic groups and the related finite groups of Lie type.

As will be seen in Part I, a linear algebraic group is an affine variety which is equipped with a group structure in such a way that the binary group operation and inversion are continuous maps. A connected (irreducible) linear algebraic group has a maximal solvable connected normal subgroup such that the quotient group is a central product of simple algebraic groups, a socalled semisimple algebraic group. Thus, one is led to the study of semisimple groups and connected solvable groups. A connected solvable linear algebraic group is the semidirect product of the normal subgroup consisting of its unipotent elements with an abelian (diagonalizable) subgroup (for example, think of the group of invertible upper triangular matrices). While one cannot expect to classify unipotent groups, remarkably enough this is possible for the semisimple quotient.

The structure theory of semisimple groups was developed in the middle of the last century and culminated in the classification of the semisimple linear algebraic groups defined over an algebraically closed field, a result essentially due to Chevalley, first made available via the Séminaire sur la classification des groupes de Lie algébriques at the Ecole Normale Supérieure in Paris, during the period 1956–1958 ([15]).