In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let be a finitely generated group. Then, for each integer define to be the number of subgroups of index in . Similarly, if is a topological group, denotes the number of open subgroups of index in . One similarly defines and to denote the number of maximal and normal subgroups of index , respectively. Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions. The theory was motivated by the desire to enumerate finite groups of given order, and the analogy with Mikhail Gromov's notion of word growth. Nilpotent groups Let be a finitely generated torsionfree nilpotent group. Then there exists a composition series with infinite cyclic factors, which induces a bijection (not though necessarily a homomorphism). such that group multiplication can be expressed by polynomial functions in these coordinates; in particular, the multiplication is definable. Using methods from the model theory of p-adic integers, F. Grunewald, D. Segal and G. Smith showed that the local zeta function is a rational function in . As an example, let be the discrete Heisenberg group. This group has a "presentation" with generators and relations Hence, elements of can be represented as triples of integers with group operation given by To each finite index subgroup of , associate the set of all "good bases" of as follows. Note that has a normal series with infinite cyclic factors. A triple is called a good basis of , if generate , and . In general, it is quite complicated to determine the set of good bases for a fixed subgroup . To overcome this difficulty, one determines the set of all good bases of all finite index subgroups, and determines how many of these belong to one given subgroup. To make this precise, one has to embed the Heisenberg group over the integers into the group over p-adic numbers. After some computations, one arrives at the formula where is the Haar measure on , denotes the p-adic absolute value and is the set of tuples of -adic integers such that is a good basis of some finite-index subgroup. The latter condition can be translated into . Now, the integral can be transformed into an iterated sum to yield where the final evaluation consists of repeated application