Corollary of Thompson transitivity theorem

Statement
Let $$G$$ be a finite group and $$p$$ is a prime number dividing the order of $$G$$. Suppose $$G$$ is a fact about::group in which every p-local subgroup is p-constrained. Suppose $$P$$ is a $$p$$-Sylow subgroup of $$G$$ and $$A$$ is maximal among abelian normal subgroups in $$P$$, with the rank of $$A$$ at least three.

Suppose $$q$$ is a prime number not equal to $$p$$.

Then:


 * 1) $$P$$ normalizes at least one maximal $$A$$-invariant $$q$$-subgroup of $$G$$.
 * 2) If $$P$$ normalizes no nontrivial $$p'$$-subgroups of $$G$$, then neither does $$A$$.

Facts used

 * 1) uses::Thompson transitivity theorem