In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let G {\displaystyle G} be a finitely generated group. Then, for each integer n {\displaystyle n} define a n ( G ) {\displaystyle a_{n}(G)} to be the number of subgroups H {\displaystyle H} of index n {\displaystyle n} in G {\displaystyle G} . Similarly, if G {\displaystyle G} is a topological group, s n ( G ) {\displaystyle s_{n}(G)} denotes the number of open subgroups U {\displaystyle U} of index n {\displaystyle n} in G {\displaystyle G} . One similarly defines m n ( G ) {\displaystyle m_{n}(G)} and s n ◃ ( G ) {\displaystyle s_{n}^{\triangleleft }(G)} to denote the number of maximal and normal subgroups of index n {\displaystyle 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 Mikhail Gromov's notion of word growth.
Nilpotent groups Let G {\displaystyle G} 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).
Z n ⟶ G {\displaystyle \mathbb {Z} ^{n}\longrightarrow G}
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
ζ G , p ( s ) = ∑ ν = 0 ∞ s p n ( G ) p − n s {\displaystyle \zeta _{G,p}(s)=\sum _{\nu =0}^{\infty }s_{p^{n}}(G)p^{-ns}}
is a rational function in p − s {\displaystyle p^{-s}} . As an example, let G {\displaystyle G} be the discrete Heisenberg group. This group has a "presentation" with generators x , y , z {\displaystyle x,\,y,\,z} and relations
[ x , y ] = z , [ x , z ] = [ y , z ] = 1. {\displaystyle [x,y]=z,[x,z]=[y,z]=1.}
Hence, elements of G {\displaystyle G} can be represented as triples ( a , b , c ) {\displaystyle (a,\,b,\,c)} of integers with group operation given by
( a , b , c ) ∘ ( a ′ , b ′ , c ′ ) = ( a + a ′ , b + b ′ , c + c ′ + a b ′ ) . {\displaystyle (a,b,c)\circ (a',b',c')=(a+a',b+b',c+c'+ab').}
To each finite index subgroup U {\displaystyle U} of G {\displaystyle G} , associate the set of all "good bases" of U {\displaystyle U} as follows. Note that G {\displaystyle G} has a normal series
G = ⟨ x , y , z ⟩ ▹ ⟨ y , z ⟩ ▹ ⟨ z ⟩ ▹ 1 {\displaystyle G=\langle x,y,z\rangle \triangleright \langle y,z\rangle \triangleright \langle z\rangle \triangleright 1}
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