A Generalization Of Glauberman-Solomon Subgroup And Its Applications

A new series of characteristic subgroups D_i(P),where i ? 1,is defined for anarbitrary finite p-group P.It also proves that if G is a p-stable group with P ? Syl_p(G),then under some nature conditions,every Oi(P)is the characteristic subgroup of G.This generalizes the main theorems of Glauberman and Solomom in[1].Furthermore,one of the applications of characteristic subgroups D_i(P)is a new criterion about p-nilpotent group,and we reduce the p-nilpotency on the finite group G to N_G(D_i(P)).In the end of this paper,we generate the above two conclusions to fusion systems.The main theorems are as follows:Theorem 1 Let G be a finite group,where p is a prime factor of |G|.Assume thatG is p-stable and C_G(O_p(G))? O_p(G).Then D_i(P)are the characteristic subgroupsof G for all i ?1.Theorem 2 Let G be a finite group,where p is an odd prime,P ? Syl_p(G).Show thatG is a p-nilpotent group if and only if for a certain i ? 1,NO(D_i(P))is a p-nilpotentgroup.We can express the version of theorem 1 and 2 in fusion systems as follows.Theorem 3 Let F be a p-stable saturated fusion on P,where P is a finite p-group.Then D_i(P)<F for i ? 1.Theorem 4 Let p be an odd prime,and let F be a saturated fusion system on afinite p-group P.Then F is a trivial fusion system if and only if N_F(D_i(P))is a trivialfusion system for a certain i>1.