Zeta function of a group

Let G be a finitely generated group and let 𝒳 be a family of finite index subgroupsMathworldPlanetmathPlanetmath of G. Define


Note that these numbers are finite since a finitely generated group has only finitely many subgroups of a given index. We define the zeta function of the family 𝒳 to be the formal Dirichlet seriesMathworldPlanetmath


Two important special cases are the zeta function counting all subgroups and the zeta function counting normal subgroupsMathworldPlanetmath. Let 𝒮(G) and 𝒩(G) be the families of all finite index subgroups of G and of all finite index normal subgroups of G, respectively. We write an(G)=an(𝒮(G)) and an(G)=an(𝒩(G)) and define




If, in addition, G is nilpotentPlanetmathPlanetmath, then ζG has a decomposition as a formal Euler productMathworldPlanetmath

ζG(s)=p primeζG,p(s),



An analogous result holds for the normal zeta function ζG. The result for both ζG and ζG can be proved using properties of the profinite completion of G. However, a simpler proof for the normal zeta function is provided by the fact that a finite nilpotent group decomposes into a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of its Sylow subgroups. These results allow the zeta functions to be expressed in terms of p-adic integrals, which can in turn be used to prove (using some high-powered machinery) that ζG,p(s) and ζG,p(s) are rational functions in p and p-s.

In the case when G is a 𝒯-group, that is, G is finitely generatedMathworldPlanetmathPlanetmath, torsion free, and nilpotent, define αG to be the abscissa of convergence of ζG. That is, αG is the smallest α such that ζG defines a holomorphic functionMathworldPlanetmath in the right half-plane {z(z)>α}. It can then be shown that αGh(G), where h(G) is the Hirsch number of G. Therefore, if G is a 𝒯-group, ζG defines a holomorphic function in some right half-plane.


  • 1 F. J. Grunewald, D. Segal, and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. math. 93 (1988), 185–223.
  • 2 M. P. F. du Sautoy, Zeta functions of groups: the quest for order versus the flight from ennui, Groups St. Andrews 2001 in Oxford. Vol. I, London Math. Soc. Lecture Note Ser., vol. 304, Cambridge Univ. Press, 2003, pp. 150–189.
Title Zeta function of a group
Canonical name ZetaFunctionOfAGroup
Date of creation 2013-03-22 15:16:00
Last modified on 2013-03-22 15:16:00
Owner avf (9497)
Last modified by avf (9497)
Numerical id 7
Author avf (9497)
Entry type Definition
Classification msc 20E07
Classification msc 20F69
Classification msc 20F18
Related topic Group