A New Characteristic P-Subgroup And Its Applications

We prove that W(P) has the same technical effect with Glauberman-Solomon subgroup D*(P) by defining a smaller characteristic subgroup W(P) and its deforma-tion We{P). Furthermore, we demonstrate a similar Glauberman-Solomon theorem: If G is a p-stable group and P is a Sylow p-subgroup of G, then W(P) is precisely a nontrivial characteristic subgroup under appropriate conditions. In order to strength-en the applications of W(P) and We(P) in finite group, we give a new criterion about G is a p-nilpotent group. That is, if Ng(W(P)) (or NG(We(P))) is p-nilpotent group, where P ∈ Sylp(G), p is an odd prime, then G is a p-nilpotent group. In addition, with the development of the fusion system, this thesis also gives an application of W(P) and its deformation We(P) in fusion system, which means that the criterion about G is a p-nilpotent group is generalized to fusion system, and demonstrates the following result:Let F be a saturated fusion system on a finite p-group P, where p is an odd prime. If NF(W(P)) (or NF>(We(P))) is trivial, then F is trivial.In this thesis, the main conclusions are as follows: Theorem 1. Let G> 1 be a finite group, where p is a prime factor of |G|, and assume the following conditions:(1) characteristic p-property:CG(Op(G))≤OP(G);(2) normal p-stability:for each normal p-subgroup P of G and g ∈ G, if [P, g, g]= 1, then ∈ Op(G), where G=G/CG(OP(G)), g=gCG(Op(G)) is the image of g in G.For any P € Sylp(G), we have that W(P) and We(P) are non-trivial characteristic subgroups of G.We have the second result immedieatedly, by using the characteristic subgroups W(P) and We(P). Theorem 2. Let G be a finite group, where p is an odd prime, P ∈ Sylp(G). Then the following are equivalent:(1) G is a p-nilpotent group;(2) NG(W(P)) is a p-nilpotent group;(3) NG(We(P)) is a p-nilpotent group.The above criterion for a finite group G to be a p-nilpotent group is generalized to fusion system. In other words, Theorem 2 can be expressed as follows by fusion system language.Theorem 3. Let F be a saturated fusion system on a finite p-group P, where p is an odd prime. Then the following are equivalent:(1) F is a trivial fusion system;(2) NF(W(P)) is a trivial fusion system;(3) NF(We(P)) is a trivial fusion system.