Fixed-point-free elements in \(p\)-groups.
type: Item
instance of: scholarly article
title: Fixed-point-free elements in \(p\)-groups.
zbMATH Open document ID: 1217.20001
author: Eleonora Crestani
author: Pablo Spiga
published in: Israel Journal of Mathematics
publication date: 2011-01-10T00:00:00Z
review text: It is known that every finite transitive permutation group \(G\) of degree \(>1\) has a fixed-point-free \(p\)-element for some prime \(p\) [see \textit{B. Fein, W. M. Kantor} and \textit{M. Schacher}, J. Reine Angew. Math. 328, 39-57 (1981; Zbl 0457.13004)]. It has been conjectured by \textit{P. J. Cameron} [Permutation groups. Cambridge: Cambridge University Press (1999; Zbl 0922.20003), Problem 5], that for each prime \(p\) and positive integer \(b\) there exists a bound \(f(p,b)\) such that any transitive permutation group of degree \(p^ab\) with \(a>f(p,b)\) has a fixed-point-free \(p\)-element. Cameron later offered a stronger conjecture: there is a bound \(F(m,p)\) such that if \(G\) is \(p\)-group which is a permutation group with \(m\) orbits, each of size at least \(p^{F(m,p)}\), then \(G\) contains a fixed-point-free element. The main result of the present paper is a counter-example to the second conjecture. The earlier conjecture remains open.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 5834350
zbMATH Keywords: fixed-point free elements of prime order
zbMATH Keywords: transitive permutation groups
MaRDI profile type: Publication
full work available at URL: s11856-010-0109-7
OpenAlex ID: W1978033620
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DOI: 10.1007/S11856-010-0109-7
description: scientific article; zbMATH DE number 5834350