# Generation of Finite Simple Groups and Derangements

 MEMBERS SEMINAR Topic: Generation of Finite Simple Groups and Derangements Speaker: Robert Guralnick Affiliation: IAS Date: Monday, April 3 Time/Room: 4:00pm - 5:00pm/S-101

We will first discuss some results on generation of finite simple groups. Using the classification of finite simple grouops, one can prove the following results: Every finite simple can be generated by two elements and the probability that a pair of random elements of a finite simple group generate the group tends to 1 as the order of the simple tends to infinity. We will discuss the relation between these results and show how in certain cases, the first implies the second. We also talk about the 3/2 generation problem -- i.e. when you specifiy the first element and ask for a generating set. Similiar ideas can be used to study derangements in finite transitive groups -- i.e. fixed point free permutations. We will discuss the problem of estimating the proportion of derangements in a finite transitive group and some related issues for algebraic groups.