On <f>p</f>-nilpotence of finite groups

All groups considered in this paper will be finite. A 2-group is called quaternion-free if it has no section isomorphic to the quaternion group of order 8. For a finite <f>p</f>-group <f>P</f> the subgroup generated by all elements of order <f>p</f> is denoted by <f>Ω1(P)</f>. Zhang [Proc. Amer. Math. Soc. 98 (4) (1986) 579] proved that if <f>P</f> is a Sylow <f>p</f>-subgroup of <f>G</f>, <f>Ω1(P)⩽Z(P)</f> and <f>NG(Z(P))</f> is <f>p</f>-nilpotent, then <f>G</f> is <f>p</f>-nilpotent, i.e., <f>G</f> has a normal Hall <f>p′</f>-subgroup. Recently, Ballester-Bolinches and Guo [J. Algebra 228 (2000) 491] proved that if <f>P</f> is a Sylow 2-subgroup <f>G</f>, <f>P</f> is quaternion-free, <f>Ω1(P∩G′)⩽Z(P)</f> and <f>NG(P)</f> is 2-nilpotent, then <f>G</f> is 2-nilpotent. Bannuscher and Tiedt [Ann. Univ. Sci. Budapest 37 (1994) 9] proved that if <f>p>2</f>, <f>P</f> is a Sylow <f>p</f>-subgroup of <f>G</f>, <f>&z.sfnc;Ω1(P∩Px)&z.sfnc;⩽pp-1</f> for all <f>x∈G&z.drule;NG(P)</f> and <f>NG(P)</f> is <f>p</f>-nilpotent, then <f>G</f> is <f>p</f>-nilpotent. The object of this paper is to improve and extend these results. [Copyright &y& Elsevier]