Probabilistic Group Theory

This survey discusses three aspects of the ways in which probability has been applied to the theory of finite groups: probabilistic statements about groups; construction of randomized algorithms in computational group theory; and application of probabilistic methods to prove deterministic theorems in group theory. It concludes with a brief summary of related results for infinite groups. Cet article donne un apercu sur trois aspects des façons dont la probabilité est appliquée à la théorie des groupes finis: les faits probabilistiques des groupes; la construction d’algorithmes aléatoires dans la computation; et l’application des moyens probabilistiques pour obtenir les theorems déterministiques dans la théorie des groupes. On termine avec un bref sommaire de resultats se rapportant aux groupes infinis. Mathematics Subject Classification 2000: 20-02, 20D60, 20F69, 20-04 In the past 20 years, and particularly during the last decade, there has been a growing interest in the use of probability in finite groups. It has been my experience that many pure mathematicians still look on probability theory as an “applied” subject (perhaps because of the way it is taught in our universities), and are dubious about the validity of using probabilistic reasoning in their own discipline. Kolmogorov’s axiomatization [51] of probability theory still seems to be a well kept secret. However, no-one should be uncomfortable in a discussion of the applications of probability theory to finite groups, since in these cases the probabilistic statements can be always be simply understood in terms of proportions. In the current article I shall consider three aspects of the ways in which probability has been applied to problems in group theory. These are: probabilistic statements about groups which give some alternative description of the structure of the group and its elements (Sects. 1 and 2); applications of probability to construct algorithms in computational group theory (Sect 3); and applications of probabilistic methods to prove deterministic theorems in group theory (see Sect. 4). The last section (Sect. 5) deals with some related results in infinite groups. Since my focus is on group theory, I shall ignore several very important areas where the primary interest is in probability theory such as random walks