Alfréd Rényi Institute of Mathematics, Budapest

**Finite simple groups**

The Classification of Finite Simple Groups is a theorem that has by far the longest proof in mathematics, its proof is scattered in numerous papers that add up to more than 10,000 journal pages. The statement is that all finite simple groups are known. Simple groups are the building blocks of finite groups, and they are known both in the technical sense that there is a list containing all isomorphism types of finite simple groups, and in the informal sense that if a general question about groups can be reduced to a problem about simple groups then a solution often can be obtained just by going through the list of finite simple groups.

In the talk I will sketch this list, that involves 18 infinite series and 26 individual (so-called poradic') groups. I will highlight some of these groups. As the 2008 Abel Prize has been awarded to two groups theorists, John G. Thompson and Jacques Tits, I should not miss mentioning the Thompson group (one of the sporadic groups) and the Tits group (an exceptional group in one of the infinite series).

I will briefly discuss the reliability of such an extremely complex proof. Some consequences of the Classification Theorem will be presented as well.