In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.
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Let G be a finitely generated group. Then, for each integer n define n(G) to be the number of subgroups U of index n in G. Similarly, if G is a topological group, s_n(G) denotes the number of open subgroups U of index n in G. One similarly defines m_n(G) and
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 G 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).
Zn → Gsuch 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 p−s.
As an example, let G be the discrete Heisenberg group. This group has a "presentation" with generators x, y, z and relations
Hence, elements of G can be represented as triples (a, b, c) of integers with group operation given by
To each finite index subgroup U of G, associate the set of all ``good bases´´ of U as follows. Note that G has a normal series
with infinite cyclic factors. A triple (g_1, g_2, g_3) in G is called a good basis of U, if g_1, g_2, g_3 generate U, and
where μ is the Haar measure on Zp,
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. From this we deduce that ζG(s) 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 ζG(s). The local factor
can always be expressed as a definable p-adic integral. Applying a result of MacIntyre on the model theory of p-adic integers, one deduces again that ζG(s) is a rational function in p−s. 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
where α is the abscissa of convergence of ζG(s), and δ is some positive number, and holomorphic in some neighbourhood of
for some real number α and a non-negative integer k.
Subgroup growth and coset representations
Let G be a group, U a subgroup of index n. Then G acts on the set of left cosets of U in G by left shift:
In this way, U induces a homomorphism of G into the symmetric group on
the stabilizer of the point 1 is a subgroup of index n in G. Since the set
can be permuted in
ways, we find that
where
In several instances the function
As an example, let
that is
From this we deduce
For more complicated examples, the estimation of