Publications of Paul Erdos

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(n^1-\epsilon) for every \epsilon > 0 when n is square-free,

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

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

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

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

Books Problems Set Theory Combinatorics Extremal Probl/Ramsey Th.

Graph Theory Add.Number Theory Mult.Number Theory Analysis Geometry

Probabability Personalia About Paul Erdös Publication Year Home Page

Books	Problems	Set Theory	Combinatorics	Extremal Probl/Ramsey Th.
Graph Theory	Add.Number Theory	Mult.Number Theory	Analysis	Geometry
Probabability	Personalia	About Paul Erdös	Publication Year	Home Page