Groups generated by two \(p\)-cycles
type: Item
instance of: scholarly article
title: Groups generated by two \(p\)-cycles
zbMATH Open document ID: 0619.20002
DOI: 10.1007/BF01196498
author: Karen R. Fawcett
author: G. L. Walls
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: The main result of this paper is a characterization of all permutation groups which can be generated by two \(p\)-cycles, \(p\) a prime. Let \(x\) and \(y\) be two \(p\)-cycles which together move \(n\) distinct points. Then either \(<x,y>=A_n\) or one of the following must occur: (i) \(n=p - <x,y>\cong Z_ p\), \(p=(r^ m-1)/(r-1)\) for some prime \(r\) and \(<x,y>\cong L_m(r)\), \(p=11\) and \(<x,y>\cong L_2(11)\) or \(M_{11}\), or \(p=23\) and \(<x,y>\cong M_{23}\), (ii) \(n=p+1 - <x,y>\cong L_2(p)\), \(p=11\) and \(<x,y>\cong M_{11}\), \(M_{12}\), or \(L_2(11)\), \(p=23\) and \(<x,y>\cong M_{24}\), or \(p\) is a Mersenne prime and \(<x,y>\) is a certain Frobenius group or a certain split extension of an elementary abelian \(2\)-group by \(L_m(2)\) where \(2^m=n\), (iii) \(n=3 - <x,y>\cong S_3\). The proof uses the characterization of simple 2-transitive groups due to \textit{P. J. Cameron} [Bull. Lond. Math. Soc. 13, 1--22 (1981; Zbl 0463.20003)].
Mathematics Subject Classification ID: 20B10
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 4004444
zbMATH Keywords: permutation groups
zbMATH Keywords: p-cycles
zbMATH Keywords: Frobenius group
zbMATH Keywords: split extension
zbMATH Keywords: simple 2- transitive groups
MaRDI profile type: Publication
cites work: Finite Permutation Groups and Finite Simple Groups
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full work available at URL: https://doi.org/10.1007/bf01196498
OpenAlex ID: W2317060934
description: scientific article; zbMATH DE number 4004444