Overview

The Classification of Finite Simple Groups was one of the biggest achievements in 20th century group theory. By Jordan Hölder theorem, every finite group has a composition series consisting of finite simple groups. So, finite simple groups can be seen as building blocks for all finite groups. According to the classification theorem, a finite non-abelian simple group is either an alternating group, or a simple group of Lie type or one of the 26 sporadic groups which are not belong to any infinite family. In this seminar, the theory of simple groups of Lie type as groups automorphisms of simple Lie algebras will be discussed from the basics. The classical simple groups will be introduced and then root systems and Weyl groups, Chevalley groups and their structure will be investigated. After dealing with automorphisms of Chevalley groups, at the end of the seminar, we will introduce some geometric structures associated with simple groups of Lie type.

Textbook: Simple Groups of Lie Type, Roger Carter, Pure and Applied Mathematics Volume XXVIII, 1972

Seminar plan: A detailed plan of the talks will be available after the first meeting. 1st meeting : The Classical Simple Groups will be introduced (K. Ersoy)

2nd meeting : Weyl groups and root system. Abstract root systems, fundamental systems, length function, definitions by generators and relations and chamber systems.

3rd meeting: Simple Lie algebras, subalgebras, Cartan decomposition, roots of a simple Lie algebra, Dynkin diagrams, descriptions of simple Lie algebras, existence and isomorphism theorems

4th meeting : Chevalley groups, Chevalley basis, structure constants, the exponential map, algebras over arbitrary fields, the groups A_1(K)

5th meeting : Unipotent subgroups, Chevalley commutator formula.

6th meeting: Root SL(2,K) subgroups, their homomorphisms

The rest of the plan will be announced after first meeting.