Probabilistic Methods in Group Theory

The application of probabilistic methods to another chapter of mathematics (number theory, different branches of analysis, graph theory etc .) has in the last 30 years often led to interesting results, which could not be obtained by the usual methods of the chapters in question . These results are in most cases of the following character : it is shown that in some sense "most" elements of a set of mathematical objects possess a certain property . Thus these results deal with typical properties of elements of a certain set, neglecting elements which behave in a different way, provided that their number is in some sense negligibly small . If the number of elements of the set considered is infinite a certain natural measure has to be introduced and relations are studied which are valid for "almostall" elements of the set considered, i .e . with the exception of subset of measure zero according to the chosen measure . If finite systems are studied, then the most natural measure is the number of elements of the sets considered . In such cases the point of view usually adopted is as follows : if S,; (n = 1, 2, • • •) is a sequence of finite sets, the number of elements of S„ being equal to N(n) where lim N(n) = + ca, and if A(n) denotes the number n-++U