Glauberman's replacement theorem

Statement
Suppose $$p$$ is an odd prime, and $$P$$ is a $$p$$-group. Let $$\mathcal{A}(P)$$ be the set of abelian subgroups of maximum order in $$P$$ and $$J(P)$$ be the  join of abelian subgroups of maximum order: the subgroup of $$P$$ generated by the members of $$\mathcal{A}(P)$$.

Suppose $$B$$ is a class two normal subgroup of $$P$$ such that its  derived subgroup is contained in the center of $$J(P)$$ (this center is also called the  ZJ-subgroup of $$P$$) in symbols:

$$[B,B] \le Z(J(P))$$.

If $$A \in \mathcal{A}(P)$$ is such that $$B$$ does not normalize $$A$$, there exists $$A^* \in \mathcal{A}(P)$$ such that:


 * $$A \cap B$$ is a proper subgroup of $$A^* \cap B$$.
 * $$A^*$$ normalizes $$A$$.

Other replacement theorems

 * Thompson's replacement theorem for abelian subgroups
 * Thompson's replacement theorem for elementary abelian subgroups

For a complete list of replacement theorems, refer:

Category:Replacement theorems

Applications

 * Any class two normal subgroup whose derived subgroup is in the ZJ-subgroup normalizes an abelian subgroup of maximum order
 * Glauberman's theorem on intersection with the ZJ-subgroup
 * p-constrained and p-stable implies Glauberman type for odd p
 * Glauberman-Thompson normal p-complement theorem