A Class Of Characteristic Subgroups Of Finite P-Groups And Its Applications

Given an arbitary finite p-group P and a positive integer i,we define a series of characteristic subgroups W_i(P)of P which have similar properties to the special subgroup Z(J(P))of P,i.e.,these W_i(P)are nontrivial characteristic subgroups of G under some suitable conditions.Moreover,we also give some applications of W_i(P)in finite groups,that is,a group G is p-nilpotent if and only if N_G(W_i(P))are p-nilpotent groups for all i.In addition an application and generalization of W_i(P)is given in saturated fiusion systems.For example,this conclusion is generalized to fusion systems.It tells us that F is trivial if and only if NF(W_i(P))are trivial.Furthermore,in order to weaken the condition of Glaubermans ZJ-theorem,we give theorem D,which means that W_i(P)are normal subgroups of F under certain conditions.The first main conclusion of this paper is as follows:Theorem A.Let G be a finite group and P be a Sylow p-subgroup of G,where p is a prime.Suppose that G is p-stable and C_G(Op(G))?Op(G),then W_i(P)are characteristic subgroups of G for all i>1.The second conclusion is an application of W_i(P).Theorem B.Let G be a finite group and P be a Sylow p-subgroup of G,where p is an odd prime.Then G is a p-nilpotent group if and only if NG(W_i(P))are p-nilpotent groups for some i>1.The following is the version of Theorem B in fusion systems.Theorem C.Let F be a saturated fusion system on a finite p-group P,where p is an odd prime.Then F is a trivial fusion systems if and only if NF(W_i(P))are trivial for a certain i?1.To weaken the condition for Glauberman ZJ-theorem in fusion systems,we have:Theorem D.Let F be a saturated fusion system on a finite p-group P.Suppose that F is p-stable,then W_i(P)(?)for all i>1.