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PROFINITE GROUPS WITH MULTIPLICATIVE PROBABILISTIC ZETA FUNCTION

Published online by Cambridge University Press:  23 July 2004

E. DETOMI  and
A. LUCCHINI
E. DETOMI
Affiliation:
Dipartimento di Matematica, Università di Brescia, Via Valotti 9, 25133 Brescia, Italydetomi@ing.unibs.it, lucchini@ing.unibs.it Current address: Dipartimento di Matematica, Università di Padova, via Belzoni 7, 35131 Padova, Italydetomi@math.unipd.it
A. LUCCHINI
Affiliation:
Dipartimento di Matematica, Università di Brescia, Via Valotti 9, 25133 Brescia, Italydetomi@ing.unibs.it, lucchini@ing.unibs.it
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Abstract

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)\,{=}\,\sum_{n}{a_n}/{n^s}$ is associated, where $a_n \,{=}\,\sum_{|G:H|=n} \mu_G(H)$. It is proved that $G$ is prosoluble if and only if the sequence $\{a_n\}_{n \in \mathbb N}$ is multiplicative, that is, $a_{rs}\,{=}\,a_ra_s$ for any pair of coprime positive integers $r$ and $s$. This extends the analogous result on the probabilistic zeta function of finite groups.

Keywords

20E1820E0711M41
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 70 , Issue 1 , August 2004 , pp. 165 - 181
Copyright
The London Mathematical Society 2004

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Footnotes

This research was partially supported by MIUR (project ‘Teoria dei gruppi e applicazioni’).