A linear bound for \(\rho(n)\)
type: Item
instance of: scholarly article
title: A linear bound for \(\rho(n)\)
zbMATH Open document ID: 0859.20014
author: Thomas Michael Keller
published in: Journal of Algebra
publication date: 1997-02-17T00:00:00Z
review text: For a positive integer \(n\), denote by \(\sigma(n)\) the number of different prime divisors of \(n\). For a finite group \(G\), denote by \(\sigma(G)\) the maximum of \(\sigma(\text{ord}(g))\) over all \(g\in G\). The paper is concerned with bounding \(\sigma(\#G)\) in terms of \(\sigma(G)\) for finite solvable groups \(G\). Let \(\rho(n)\) be the maximum of \(\sigma(\#G)\) over all finite solvable groups \(G\) such that \(\sigma(G)=n\). In earlier work of the author [J. Algebra 174, No. 2, 587-598 (1995; see the preceding review Zbl 0859.20013)], it was shown that \(\rho(n)=O(n\log n)\) as \(n\to\infty\), improving the previously known quadratic bound. In the present paper, the author now obtains a linear bound on \(\rho(n)\), namely that \[ \limsup_{n\to\infty}{\rho(n)\over n}\leq 4. \] This is essentially best possible, since in an earlier paper [J. Algebra 170, No. 2, 625-648 (1994; Zbl 0816.20022)], he had shown that \[ \limsup_{n\to\infty}{\rho(n)\over n}\geq 3. \] As to the actual behaviour of \(\rho\), he conjectures that in fact \(\rho(n)=3n\) for all \(n\geq 4\). See also the author's doctoral thesis [Zbl 0859.20018].
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 840722
zbMATH Keywords: arithmetic conditions on element orders
zbMATH Keywords: numbers of prime divisors
zbMATH Keywords: finite solvable groups
reviewed by: Michael Stoll
MaRDI profile type: Publication
full work available at URL: jabr.1995.1370
OpenAlex ID: W2042192787
DOI: 10.1006/JABR.1995.1370
description: scientific article; zbMATH DE number 840722