Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  612.10038
Autor:  Erdös, Paul; Murty, M.Ram; Murty, V.Kumar (Ram Murty, M.; Kumar Murty, V.)
Title:  On the enumeration of finite groups. (In English)
Source:  J. Number Theory 25, 360-378 (1987).
Review:  Let G(n) denote the number of non-isomorphic groups of order n. Using methods of analytic number theory and an explicit algebraic equation of Hölder, the authors derive interesting asymptotic information about G(n) when n is square-free. The main results are: (i)  G(n) = \Omega(n1-\epsilon) for every \epsilon > 0 when n is square-free,

(ii)  log G(n) = (1+o(1))  log log nsump|n(log p)/(p-1) for almost all square-free n. In addition, they derive asymptotic estimates for Fk(x) where Fk(x) = card{n \leq x: G(n) = k}:

Fk(x) = (c(a)+o(1)) x/(log log log x)a+1 for k = 2a,

Fk(x) = O(x/(log log x)1-\epsilon) for k\ne 2a,

where c(a) is an appropriate constant.
Reviewer:  J.Knopfmacher
Classif.:  * 11N45 Asymptotic results on counting functions for other structures
                   20D60 Arithmetic and combinatorial problems on finite groups
Keywords:  number of non-isomorphic groups; asymptotic estimates

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