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• W169646329 abstract "In this thesis, we investigate the connection between finitely generated profinite groups Gand the associated Dirichlet series PG(s) of which the reciprocal is called the probabilisticzeta function of G. In particular, we consider the conjecture of Lucchini saying that givena finitely generated profinite group G, the associated Dirichlet series PG(s) is rational ifand only if the quotient group G/Frat(G) is finite. Detomi and Lucchini first showed thatthe conjecture holds when G is prosoluble. For non-prosoluble groups, they later showedthat the conjecture also holds when almost every nonabelian composition factor of G is analternating group. In this thesis, we prove the conjecture in several other cases. We firstshow that it holds when almost every nonabelian composition factor of G is isomorphicto a simple group of Lie type over a field of characteristic p, where p is a fixed prime.When there are different characteristics, the problem becomes quite difficult and we donot have any answer yet. However, we obtain that the conjecture holds when almost everynonabelian composition factor of G is PSL(2, p) for some prime p 5. This is also thecase when we replace PSL(2, p) by a sporadic simple group. The conjecture is still open ingeneral and it cannot be proved by our techniques. We give some examples supportingthis. Nevertheless, we also obtain a partial result by showing that the conjecture holdswhen almost every nonabelian composition factor is isomorphic to either PSL(2, p) forsome prime p 5, or a sporadic simple group, or an alternating group Alt(n) where n iseither a prime or a power of 2." @default.
• W169646329 created "2016-06-24" @default.
• W169646329 creator A5001404329 @default.
• W169646329 date "2013-05-14" @default.
• W169646329 modified "2023-09-24" @default.
• W169646329 title "Profinite groups with a rational probabilistic zeta function" @default.
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