Zeta function of a group
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
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.
- 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|
|Date of creation||2013-03-22 15:16:00|
|Last modified on||2013-03-22 15:16:00|
|Last modified by||avf (9497)|