Probabilistic group theory  pp. 648-678


By Aner Shalev

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Abstract

In recent years there have been several developments in the study of probabilistic aspects of certain finite and profinite groups, and various conjectures in this field were settled. Moreover, the probabilistic approach led to the solution of interesting problems whose formulation had nothing to do with probability; these include problems regarding the modular group, free groups, as well as conjectures on finite permutation groups. In this lecture series I will try to survey these developments and discuss directions for further research.

Contents:

  • Finite simple groups: random generation
  • Applications: free groups, the modular group, symmetric groups
  • Profinite groups I: Hausdorff dimension
  • Profinite groups II: random generation
  • Permutation groups: minimal degree, genus, base size

Finite simple groups: random generation

Group theory and measure theory seem to intersect highly non-trivially, and so there are many branches in mathematics which could be referred to as probabilistic group theory. In this lecture series I would like to focus on a relatively young area, which concerns probabilistic aspects of finite groups and their inverse limits. I shall also demonstrate how probabilistic ideas can be used to solve classical problems in finite and infinite groups.

A classical scheme, applied successfully in combinatorics, number theory, and other areas, is to prove existence theorems using a probabilistic approach. The idea is to show that most objects have a certain property, and then to deduce that an object with that property exists.