Abstract

The probabilistic method was pioneered by Paul Erdős in the 1960s. This method has seen wide applications in many areas of mathematics, such as number theory, real analysis, combinatorics and in computer science. 

In this talk, we will mainly focus on its applications in group theory. We will begin by introducing the main idea of the probabilistic method. Then we will see how this method is applied to various group theory problems. The first application concerns the generation of finite simple groups. This includes the random generation and the (2,3)-generation of finite simple groups. In the second application, we will consider the probabilistic method introduced by Liebeck and Shalev in the 1990s to prove Camerons conjecture on base sizes for almost simple primitive groups. Finally, if time permits, we will look at a new concept, the Saxl graph, related to bases for permutation groups. We will see some new results in a recent joint work with Tim Burness on the Saxl graph using the probabilistic method.