Asymptotic Group Theory, Volume 48, Number 4

W hat is the probability that two randomly chosen permutations of 1, . . . , n generate the group Sn of all permutations of 1, . . . , n ? How many subgroups of index n can a group with two generators have? What can be said about a group of order 2n which has a nonidentity commutator of length n− 10? These questions all concern different aspects of a field which is very active nowadays and which may be called asymptotic group theory. Roughly speaking, asymptotic group theory can be thought of as groups viewed from some distance. The finer details disappear, and the rough lines become the main focus. If in group theory one often aims at full classification (say, classifying finite simple groups or doubly transitive permutation groups), then in asymptotic group theory we would be happy with a classification up to finitely many (unspecified) exceptions. If in group theory one often likes to compute certain invariants, in asymptotic group theory we would be happy with finding out the asymptotic behavior of these invariants. If in group theory one often studies a single finite group, in asymptotic group theory we often study an infinite family of finite groups or sometimes the set of finite quotients of some given infinite group. The topic of asymptotic group theory is motivated by and interrelated with many fields of mathematics and other areas; these include combinatorics, computer science, probability theory, geometry, as well as some branches of logic, algebra, and analysis. It is impossible to cover all aspects of the field in this short expository paper; important aspects such as Cayley graphs and random walks on groups, word growth of groups, permutation groups, and algorithmic applications will not be discussed. We will be content with sampling recent progress in three areas and providing some references for further reading. In the first section we shall focus on random generation and the recent use of probability in proving existence theorems in finite groups. In the following section we will turn to infinite groups and to the enumeration of their finite index subgroups. The last section combines finite groups with infinite groups; it describes the so-called coclass theory for p-groups and pro-p groups, with a few tips on the more general subject of slim objects.