Normal zeta function of a group

Definition

Let $G$ be a group. The normal zeta function of $G$ is defined as:

${\zeta }_G(s)=\sum _{n=1}^{\mathrm{\infty }}{a}_n(G)n^{-s}$

where $a_n(G)$ denotes the number of normal subgroups of $G$ of index $n$. Equivalently, it is:

${\sum }_{H{\underset{\_}{◃}}_{f}G}[G:H{]}^{-s}$

summing up over all normal subgroups of finite index in $G$.

The coefficients $a_n(G)$ are all finite when the group $G$ is finitely generated. This follows from finitely generated implies finitely many homomorphisms to any finite group (see also group with finitely many homomorphisms to any finite group).

Related notions

• Zeta function of a group

References

Journal references

• Subgroups of finite index in nilpotent groups by F. J. Grunewald, D. Segal and G. C. Smith, Inventiones mathematicae, Volume 93,Number 1, Page 185 - 223(Year 1988): ^{Gated copy (PDF)More info}