The Generalization Of Burnside P-normal Theorem In Fusion Systems

The aim of this paper is to extend Burnside p-normal theorem and Frobeniusp-nilpotency theorem to fusion systems.The notion of p-normal subgroups is intro-duced for saturated fusion systems in this paper,which is used to prove the version ofBurnside p-normal theorem of fusion systems.Furthermore,the relationship betweenthe weakly closed subgroups and the p-normal subgroups is established.In particular,a useful criterion is presented for p-normal subgroups to be weakly subgroups.Finally,as an application of our theorems,a new proof for Frobenius p-nilpotency theorem isobtained in fusion systems.The main conclusions of this paper are as follows:Theorem 1 Let F be a saturated fusion system on a finite p-group P.If M(?)Pand M is not a p-normal subgroup of F,then there is a fully F-normalized,F-centric,F-radical subgroup E such that M ? E and o(?)= qn,where ?? AutF(E)and g? pis a prime,with the following properties:(1)M??M,(2)J =<M,M?,…,M?n-1?E,(3)J?= J and CJ(?)<J,i.e.,a normalizes J but not centralize J.In particular,AutF(J)is not a p-subgroup and the p-local subsystem NF(J)is nottrivial,i.e.NF(J)?FNp(Np(J)).Theorem 2 Let F be a saturated fusion system on a finite p-group P.If F is trivial,then every subgroup of F is a p-normal subgroup.Theorem 3 Let F be a saturated fusion system on a finite p-group P.If M isa characteristic subgroup of P and every subgroup in its F-conjugacy is a normalsubgroup,then M is a weakly F-closed subgroup.