Subgroup growth

Im mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.

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 $s_{n}^{\triangleleft }(G)$ to denote the number of maximal and normal subgroups of index n, 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 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 $\mathbb{Z}^n\rightarrow G$ (which is not a homomorphism!). The 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

$\zeta _{G,p}(s)=\sum _{\nu =0}^{\infty }s_{p^{n}}(G)p^{-ns}$

is a rational function in $p^{-s}$.

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:

$g(hU)=(gh)U$.

In this way, U induces a homomorphism of G into the symmetric group on $G/U$. G acts transitively on $G/U$, and vice versa, given a transitive action of G on

$\{1,\ldots ,n\}$,

the stabilizer of the point 1 is a subgroup of index $n$ in G. Since the set

$\{2,\ldots ,n\}$

can be permuted in

$(n-1)!$

ways, we find that $s_{n}(G)$ is equal to the number of transitive G-actions divided by $(n-1)!$. Among all G-actions, we can distinguish transitive actions by a sifting argument, to arrive at the following formula

$s_{n}(G)={\frac {h_{n}(G)}{(n-1)!}}-\sum _{nu=1}^{n-1}{\frac {h_{n-\nu }(G)s_{\nu }(G)}{(n-\nu )!}},$

where $h_{n}(G)$ denotes the number of homomorphisms

$\varphi :G\rightarrow S_{n}<math>.Inseveralinstancesthefunction<math>h_{n}(G)$ is easier to be approached then $s_{n}(G)$, and, if $h_{n}(G)$ grows sufficiently large, the sum is of negligible order of magnitude, hence, one obtains an asymptotic formula for $s_{n}(G)$.

As an example, let $F_{2}$ be the free group on two generators. Then every map of the generators of $F_{2}$ extends to a homomorphism

$F_{2}\rightarrow S_{n}<math>,thatis,:<math>h_{n}(F_{2})=(n!)^{2}<math>.Fromthiswededuce:<math>s_{n}(F_{2})\sim n\cdot n!<math>.Formorecomplicatedexamples,theestimationof<math>h_{n}(G)<math>involvesthe[[representationtheory]]and[[statisticalproperties]]ofsymmetricgroups.[[Category:Grouptheory]]$