Every p-subgroup is well-placed in some Sylow subgroup

Statement that involves choosing a Sylow subgroup
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$W$$ is a fact about::conjugacy functor on the $$p$$-subgroups of $$G$$. Suppose $$H$$ is a $$p$$-subgroup of $$G$$. Then, there exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ such that $$H$$ is a fact about::well-placed subgroup in $$P$$ relative to the conjugacy functor $$W$$.

Statement that involves conjugating the subgroup
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$W$$ is a fact about::conjugacy functor on the $$p$$-subgroups of $$G$$. Suppose $$H$$ is a $$p$$-subgroup of $$G$$ contained in a $$p$$-Sylow subgroup $$Q$$. Then, there exists a subgroup $$K$$ of $$G$$, that is well-placed in $$Q$$, and is conjugate to $$H$$ in $$G$$.

Related facts

 * Every p-subgroup is conjugate to a p-subgroup whose normalizer in the Sylow is Sylow in its normalizer

Facts used

 * 1) uses::Conjugacy functor sends every subgroup to a normalizer-relatively normal subgroup: For any $$p$$-subgroup $$H$$, $$H \le N_G(W(H))$$.
 * 2) uses::Sylow subgroups exist
 * 3) uses::Sylow implies order-dominating

Proof
Given: A finite group $$G$$, a prime $$p$$, a conjugacy functor $$W$$ on the $$p$$-subgroups. A $$p$$-subgroup $$H$$ of $$G$$.

To prove: There exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ such that $$H$$ is well-placed in $$G$$ relative to $$W$$.

Proof: Define the following ascending chain of subgroups $$H_i$$. $$H_0 = H$$, and $$H_{i+1}$$ is a $$p$$-Sylow subgroup of $$N_G(W(H_i))$$. By fact (1), this chain of subgroups is an ascending chain of $$p$$-subgroups. Let $$P$$ be a $$p$$-Sylow subgroup of $$G$$ containing the union of the $$H_i$$s (such a $$P$$ exists by facts (2) and (3)). We now prove that $$H$$ is well-placed in $$P$$. This follows easily from the definition.