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Journal of the London Mathematical Society 2004 70(1):165-181; doi:10.1112/S0024610704005307
© 2004 by London Mathematical Society
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© The London Mathematical Society

Profinite Groups with Multiplicative Probabilistic Zeta Function

E. Detomi and A. Lucchini

Dipartimento di Matematica, Università di Brescia Via Valotti 9, 25133 Brescia, Italy, detomi{at}ing.unibs.it, lucchini{at}ing.unibs.it

Received 28 April 2003. Revision received 22 October 2003.

To a finitely generated profinite group G, a formal Dirichlet series PG(s)={sum}nan/ns is associated, where an = {sum}|G:H|=n µG(H). It is proved that G is prosoluble if and only if the sequence {an}nisinN is multiplicative, that is, ars = aras for any pair of coprime positive integers r and s. This extends the analogous result on the probabilistic zeta function of finite groups.

Dipartimento di Matematica, Università di Padova, via Belzoni 7, 35131 Padova, Italy, detomi{at}math.unipd.it


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