Lemma on containment in p'-core for Thompson transitivity theorem

Statement
Suppose $$G$$ is a finite group that is a group in which every p-local subgroup is p-constrained. Suppose $$A$$ is maximal among abelian normal subgroups in a $$p$$-Sylow subgroup of $$G$$, and $$A$$ is not cyclic, i.e., $$A \in SCN_2(p)$$.

Suppose $$q$$ is a prime number distinct from $$p$$, and $$Q$$ is an $$A$$-invariant $$q$$-subgroup of $$G$$. Suppose $$H$$ is a subgroup of $$G$$ for which the p-core $$O_p(H)$$ is nontrivial, and $$AQ \le H$$, then $$Q \le O_{p'}(H)$$.

Applications

 * Thompson transitivity theorem

Facts used

 * 1) uses::Maximal among abelian normal subgroups in p-Sylow subgroup that is not cyclic implies every invariant p'-subgroup is in the p'-core in p-constrained group