ON WEAKLY -SUPPLEMENTED SUBGROUPS OF SYLOW p-SUBGROUPS OF FINITE GROUPS

摘要

A subgroup H is called weakly M-supplemented in a finite group Gnif there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is anmaximal subgroup of H/HG, then H1B = BH1 < G,where HG is the largest normalnsubgroup of G contained in H. In this paper we will prove the following: Let G be anfinite group and P be a Sylow p-subgroup of G,where p is the smallest prime divisornof |G|. Suppose that P has a non-trivial proper subgroup D such that all subgroupsnE of P with order |D| and 2|D| (if P is a non-abelian 2-group, |P : D| > 2 and therenexists D1 u0002 E ≤ P with 2|D1|=|D| and E/D1 is cyclic of order 4) have p-nilpotentnsupplement or weakM-supplement in G, then G is p-nilpotent.