On groups of squarefree order
type: Item
instance of: scholarly article
title: On groups of squarefree order
zbMATH Open document ID: 0531.10048
DOI: 10.1007/BF01456092
author: M. Ram Murty
author: V. Kumar Murty
published in: Mathematische Annalen
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163896
review text: Continuing their interesting investigations of enumeration problems in finite group theory [see J. Number Theory 17, 29--36 (1983; Zbl 0516.10040), and the preceding review], the authors derive an exact formula for the number \(C(n)\) of groups of order \(n\) whose Sylow subgroups are all cyclic. This is used to prove a formula of Hölder for the total number \(G(n)\) of groups of square-free order \(n\), and then to derive the asymptotic formulae \[ \sum_{n\leq x}\log C(n) \sim Ax \log \log x;\quad \sum_{n\leq x}\mu^ 2(n) \log G(n) \sim Cx \log \log x \] as \(x\to \infty\), where \(A,C\) are explicitly stated constants. As may be expected, the arguments used involve both detailed group theory, and methods and results of analytic number theory.
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 3841987
zbMATH Keywords: number of non-isomorphic groups
zbMATH Keywords: cyclic Sylow subgroups
zbMATH Keywords: enumeration problems
zbMATH Keywords: square-free order
reviewed by: John Knopfmacher
MaRDI profile type: Publication
cites work: Q3253828
cites work: On the number of groups of a given order
cites work: AN ENUMERATION THEOREM FOR FINITE GROUPS
OpenAlex ID: W2053848351
description: scientific article; zbMATH DE number 3841987