Probabilistic Methods

Abstract

Probability has entered into group theory along several paths. One, initiated by Erdos and Turan in the 1960s, is the investigation of probabilistic properties of finite groups; a typical example of this is the classic theorem of Dixon that a random pair of elements generates the alternating group Alt(n) with probability that tends to 1 as n → ∞, and its recent extension by Kantor, Lubotzky, Liebeck and Shalev to the definitive result: a random pair of elements generates a finite simple group with probability that tends to 1 as the group order tends to ∞. Another path is based on the fact that a profinite group G, being a compact topological group, has a finite Haar measure μ. Normalizing this so that μ(G) = 1, we may consider G as a probability space: this means that the measure of a subset X of G is construed as the probability that a random element of G lies in X. It is now natural to ask questions such as: what is the probability that a random k-tuple of elements generates G? Formally, this probability is defined as: P(G,k)=

$$\mu \left\{ {({x_1}, \ldots ,{x_k}) \in {G^{(k)}}|\left\langle {{x_1}, \ldots ,{x_k}} \right\rangle = G} \right\},$$

(1)

(11.1) where μ denotes also the product measure on G_(k).