Glauberman-Solomon Subgroups And P-Nilpotent Groups

Glanberman and Solomon defined D*(P) and D*(P), in 2012. Using the two definition, we give a new criterion about a finite group G which is a p-nilpotent group: Let p be an odd prime, then the group G is a p-nilpotent group if and only if NG (D* (P)) is a p-nilpotent group, and also, if and only if NG(De*(P)) is a p-nilpotent group. What’s more we give an application of the theorem.The main conclusions of this thesis are as follows:Theorem 1. Let G be a finite group and p be a prime. Suppose that P is a Sylow p-subgroup of G. Then the following are equivalent:(1) G is a p-nilpotent group.(2) NG(D*(P)) is a p-nilpotent group.(3) NG(De*(P)) is a p-nilpotent group.The following is the version in fusion systems of the above Theorem 1.Theorem 2. Let F be saturated fusion systems of finite p-group on P, p be an odd prime. Then the following are equivalent:(1) F is trivial.(2) NF(D*(P)) is trivial.(3) NF(D*e(P)) is trivial.