Quasi-central elements and p-nilpotence of finite groups

• Autores: Huaquan Wei, Yanming Wang, Guohua Qian
• Localización: Publicationes Mathematicae Debrecen, ISSN 0033-3883, Tomus 77, Fasc. 1-2, 2010, págs. 233-244
• Idioma: inglés
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• Resumen

  • Let G be a ¯nite group and let P be a Sylow p-subgroup of G. An element x of G is called quasi-central in G if hxihyi = hyihxi for each y 2 G. In this paper, it is proved that G is p-nilpotent if and only if NG(P) is p-nilpotent and, for all x 2 GnNG(P), one of the following conditions holds: (a) every element of P \ Px \ GNp of order p or 4 is quasi-central in P; (b) every element of P \ Px \ GNp of order p is quasi- central in P and, if p = 2, P \ Px \ GNp is quaternion-free; (c) every element of P \ Px \GNp of order p is quasi-central in P and, if p = 2, [­2(P \ Px \GNp ); P] · Z(P \ GNp ); (d) every element of P \ GNp of order p is quasi-central in P and, if p = 2, [­2(P \ Px \ GNp ); P] · ­1(P \ GNp ); (e) j­1(P \ Px \ GNp )j · pp¡1 and, if p = 2, P \ Px \ GNp is quaternion-free; (f) j­(P \ Px \ GNp )j · pp¡1. That will extend and improve some known related results.