Thompson's second normal p-complement theorem

Statement
Suppose $$p$$ is an odd prime number and $$G$$ is a fact about::strongly p-solvable group that is also p-core-free. Suppose $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Then:

$$G = C_G(Z(P))N_G(J^*(P))$$

where $$Z(P)$$ is the center of $$P$$, $$C_G$$ denotes the centralizer, $$J^*(P)$$ is the join of abelian subgroups of maximum rank, and $$N_G$$ denotes the normalizer.

Related facts

 * Glauberman-Thompson normal p-complement theorem