P-constrained and p-stable implies normalizer of D*-subgroup generates whole group with p'-core for odd p

History
This statement was proved in a joint paper of Glauberman and Solomon.

General statement
Suppose $$G$$ is a finite group and $$p$$ is an odd prime number. If $$G$$ is both p-constrained and  p-stable, then the  D*-subgroup functor is a  characteristic p-functor whose normalizer generates whole group with p'-core.

Statement for p'-core-free finite groups
Suppose $$G$$ is a finite group and $$p$$ is an odd prime number. Suppose $$O_{p'}(G)$$ is trivial, i.e., $$G$$ has no nontrivial normal $$p'$$-subgroup. If $$G$$ is both p-constrained and  p-stable, then the following equivalent conditions are satisfied:


 * 1) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, $$D^*(P)$$ is a normal subgroup of $$G$$.
 * 2) For one (and hence every) $$p$$-Sylow subgroup $$P$$ of $$G$$, $$D^*(P)$$ is a characteristic subgroup of $$G$$.

Applications

 * Strongly p-solvable implies normalizer of D*-subgroup generates whole group with p'-core for odd p