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Subgroup growth and sieve methods
Published online by Cambridge University Press: 01 March 1997
Abstract
We study the subgroup growth of profinite groups. We obtain a structure theorem for profinite groups of polynomial subgroup growth (PSG groups, for short), which essentially reduces their characterization to the case where the group is a cartesian product of finite simple groups. Analysing the growth behaviour of such cartesian products, we construct, for any real number $\alpha \ge 1$, a PSG profinite group whose degree is exactly $\alpha$. Applications to the behaviour of the abscissa of convergence of the associated zeta function $\sum a_n(G)n^{-s}$ are drawn. We also show that there is no gap between polynomial and non-polynomial subgroup growth by constructing non-PSG groups whose subgroup growth is arbitrarily slow. Our arguments rely heavily on the use of sieve methods in number theory. In particular, a Bombieri-type short intervals theorem and the so-called Fundamental lemma in sieve theory play an essential role in this paper.
1991 Mathematics Subject Classification: 20E07, 11N36.
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- Research Article
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- London Mathematical Society 1997
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