关于p幂零群的几点注记

SEVERAL REMARKS FOR p-NILPOTENT GROUP

设P是有限群G的一个Sylow p子群.令1<Z_1(P)=Z(P)<Z_2(P)<Z_3(P)<…<Z_n(P)=P是P的上中心列.如果各个Z_i(P)在P中弱闭(关于G),则说P的上中心列是弱闭的.本文使用初等方法证明了下述两个结果:(1)G是P幂零的(?)P的上中心列是弱闭的且N_G(p)是p幂零的;(2)设对任意子群Q≤P,当Z(P)≤Q时,N_G(Q)/C_G(Q)是p群,则P的上中心列是弱闭的.由(1)和(2)直接得到7个推论,其中包括一批已知的正规p补定理.

Let P be a Sylow p-Subgroup of a finite group G. If every term Z,(P) in the upper central series of P is weakly closed in P with respect to G, then the upper central series of P is said to be weakly closed. By a elementary procedure author proves: (l) G is p-nilpotent if and only if the upper central series of P is weakly closed and N (P) is p-nilpotent: (2) If N (Q)/CG(Q) is a p-group for every Q with Z(P)<Q<P, then the upper central series of P is weakly closed. From (1) and (2) follow directly seven corollaries and some of them are well-known theorems about normal p-complement.