In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.
to denote the number of maximal and normal subgroups of index
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.
be a finitely generated torsionfree nilpotent group.
Then there exists a composition series with infinite cyclic factors, which induces a bijection (though not necessarily a homomorphism).
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
of integers with group operation given by To each finite index subgroup
has a normal series with infinite cyclic factors.
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.
-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 of the formula for the value of the geometric series.
can be expressed in terms of the Riemann zeta function as For more complicated examples, the computations become difficult, and in general one cannot expect a closed expression for
The local factor can always be expressed as a definable
Applying a result of MacIntyre on the model theory of
Moreover, M. du Sautoy and F. Grunewald showed that the integral can be approximated by Artin L-functions.
Using the fact that Artin L-functions are holomorphic in a neighbourhood of the line
, they showed that for any torsionfree nilpotent group, the function
is some positive number, and holomorphic in some neighbourhood of
Using a Tauberian theorem this implies for some real number
acts on the set of left cosets of
, and vice versa, given a transitive action of
on the stabilizer of the point 1 is a subgroup of index
Since the set can be permuted in ways, we find that
-actions, we can distinguish transitive actions by a sifting argument, to arrive at the following formula where
denotes the number of homomorphisms In several instances the function
grows sufficiently large, the sum is of negligible order of magnitude, hence, one obtains an asymptotic formula for
extends to a homomorphism that is From this we deduce For more complicated examples, the estimation of
involves the representation theory and statistical properties of symmetric groups.