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Element orders and Sylow structure of finite groups

Element orders and Sylow structure of finite groups Math. Z. 252, 223–230 (2006) Mathematische Zeitschrift DOI: 10.1007/s00209-005-0856-z Element orders and Sylow structure of finite groups 1 2 2 Gunter Malle , Alexander Moreto ´ , Gabriel Navarro FB Mathematik, Universitat ¨ Kaiserslautern, Postfach 3049, D-67653 Kaiserslautern, Germany (e-mail: alle@mathematik.uni-kl.de) Departament d’Algebra, Universitat de Valencia, ` 46100 Burjassot, Valencia, ` Spain (e-mail: Alexander.Moreto@uv.es; gabriel@uv.es) Received: 15 March 2005 / Published online: 16 August 2005 – © Springer-Verlag 2005 1. Introduction N. Chigira, N. Iiyori and H. Yamaki proved in the Main Theorem of their Inventi- ones paper [4] that if a finite group of even order G does not have any elements of order 2p, p a prime, then the Sylow p-subgroups of G are abelian. The main result of this paper is an attempt to extend that theorem to odd primes. Recall that given a set of primes π , O (G) is the largest normal π -subgroup of G. Theorem A. Let G be a finite group and p = q prime integers. If G does not have any elements of order pq , then one of the following holds: (i) The Sylow p-subgroups or the Sylow q -subgroups of G are abelian. (ii) G/O  http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Element orders and Sylow structure of finite groups

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Publisher
Springer Journals
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-005-0856-z
Publisher site
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Abstract

Math. Z. 252, 223–230 (2006) Mathematische Zeitschrift DOI: 10.1007/s00209-005-0856-z Element orders and Sylow structure of finite groups 1 2 2 Gunter Malle , Alexander Moreto ´ , Gabriel Navarro FB Mathematik, Universitat ¨ Kaiserslautern, Postfach 3049, D-67653 Kaiserslautern, Germany (e-mail: alle@mathematik.uni-kl.de) Departament d’Algebra, Universitat de Valencia, ` 46100 Burjassot, Valencia, ` Spain (e-mail: Alexander.Moreto@uv.es; gabriel@uv.es) Received: 15 March 2005 / Published online: 16 August 2005 – © Springer-Verlag 2005 1. Introduction N. Chigira, N. Iiyori and H. Yamaki proved in the Main Theorem of their Inventi- ones paper [4] that if a finite group of even order G does not have any elements of order 2p, p a prime, then the Sylow p-subgroups of G are abelian. The main result of this paper is an attempt to extend that theorem to odd primes. Recall that given a set of primes π , O (G) is the largest normal π -subgroup of G. Theorem A. Let G be a finite group and p = q prime integers. If G does not have any elements of order pq , then one of the following holds: (i) The Sylow p-subgroups or the Sylow q -subgroups of G are abelian. (ii) G/O 

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Mathematische Zeitschrift – Springer Journals

Published: Aug 16, 2005

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