You are here
Home ›Zeta function of a group
Primary tabs
Zeta function of a group
Let be a finitely generated group and let be a family of finite index subgroups of . 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 to be the formal Dirichlet series
Two important special cases are the zeta function counting all subgroups and the zeta function counting normal subgroups. Let and be the families of all finite index subgroups of and of all finite index normal subgroups of , respectively. We write and and define
and
If, in addition, is nilpotent, then has a decomposition as a formal Euler product
where
An analogous result holds for the normal zeta function . The result for both and can be proved using properties of the profinite completion of . However, a simpler proof for the normal zeta function is provided by the fact that a finite nilpotent group decomposes into a direct product of its Sylow subgroups. These results allow the zeta functions to be expressed in terms of -adic integrals, which can in turn be used to prove (using some high-powered machinery) that and are rational functions in and .
In the case when is a -group, that is, is finitely generated, torsion free, and nilpotent, define to be the abscissa of convergence of . That is, is the smallest such that defines a holomorphic function in the right half-plane . It can then be shown that , where is the Hirsch number of . Therefore, if is a -group, defines a holomorphic function in some right half-plane.
References
- 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.
Mathematics Subject Classification
20E07 Subgroup theorems; subgroup growth20F69 Asymptotic properties of groups
20F18 Nilpotent groups
- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)
- Other useful stuff
Recent Activity
new question: Sorry to steal a few minutes of your time for this question, but i honestly don't know what else to do. by Whrazithar
new question: equality of the determinants of submatrices of an orthogonal matrix by ismayli
Jun 11
new correction: Typo by suitangi
Jun 2
new question: Creating another set with same cardinality. by hkkass
Jun 1
new image: ProblemOneRevised by unlord
new Education: Chapter II by rspuzio
May 31
new collection: The Calculus by Davis and Brenke by rspuzio
new question: Proofs by weixifan
new question: Summation Integration Question by trevor.nickle
May 27
new correction: typo+finite measure hypothesis by Filipe


