On the enumeration of finite groups
type: Item
instance of: scholarly article
title: On the enumeration of finite groups
zbMATH Open document ID: 0612.10038
DOI: 10.1016/0022-314X(87)90039-4
author: M. Ram Murty
author: V. Kumar Murty
author: P. Erdős
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: 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: \[ G(n) = \Omega(n^{1-\varepsilon})\text{ for every \(\varepsilon > 0\) when \(n\) is square-free,} \tag{i} \] \[ \log G(n) = (1+o(1))\;\log\log n\sum_{p\mid n}(\log p)/(p-1) \tag{ii} \] for almost all square-free \(n\). In addition, they derive asymptotic estimates for \(F_k(x)\) where \(F_k(x)=\text{card}\{n\leq x: G(n)=k\}\): \[ F_k(x) = (c(a)+o(1)) x/(\log\log\log x)^{a+1}\text{ for } k = 2^a, \] \[ F_k(x) = O(x/(\log\log x)^{1-\varepsilon})\text{ for } k\neq 2^a, \] where \(c(a)\) is an appropriate constant.
reviewed by: John Knopfmacher
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 3989466
zbMATH Keywords: number of non-isomorphic groups
zbMATH Keywords: asymptotic estimates
Wikidata QID: Q104036881
MaRDI profile type: Publication
cites work: Q3867876
cites work: Q5802214
cites work: Q4052222
cites work: On the number of groups of a given order
cites work: On groups of squarefree order
cites work: AN ENUMERATION THEOREM FOR FINITE GROUPS
full work available at URL: 0022-314x(87)90039-4
OpenAlex ID: W2055475712
description: scientific article; zbMATH DE number 3989466