p-拟正规子群

P-QUASI-NORMAL SUBGROUPS

本文引进了 p-拟正规子群的概念,讨论了 p-拟正规子群对群结构的影响,主要结果有:(1) G 的极大子群均 p-拟正规■Gp-闭;(2) G 的2-极大子群均 p-拟正规■Gp-闭或 G 为有指数为 p 的循环正规子群的 p~αq 阶亚循环群,p~α|q-1;(3) 若 G 有一循环极大子群 p-拟正规,则 G 超可解或 G 可解且 p-闭;(4) ■ p||G|,G 的 Sylow p-子群的所有极大子群均 p-拟正规,则 G=F_0又 F_1,其中 F_0为G 的幂零正规的 Hall 子群,F_1是 Sylow 子群全循环的群.

This paper introduces the concept of p-quasl-normal subgroup,and investigates how the structure of a group Gis influenced by the p-quasi-normal subgroups of G.We obtains the following theorems:(1) every maximal subgroup of G is p-quasi-normal in G if and only it G is p-quasi-normal in G if and only if G is p-closed;(2) every 2-maximal subgroup of G is p-quasi-normal in G if and only if G is p-closed or G is a group of order p^aq,P~|q-1,whose every Sylow subgroup is cycle,and which has a cycle normal subgroup whose index is p;(3) if G has a maximal subgroup which is cycle and p-quasi-normal,then G is supersolvable or solv-able and p-closed;(4) (?) Pi||G|,all maximal subgroup of every Sylow subgroup of G is p-quasi-normal in G,then G=F_0(?)F_1, F_0 is the nilpotent normal Hall·subgroup of G,F_1 is a subgroup of G whose every Sylow subgroup is cycle.