Well-placed subgroup

Definition
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Suppose $$W$$ is a defining ingredient::conjugacy functor on $$G$$ with respect to the prime $$p$$.

A subgroup $$H$$ of $$P$$ is termed well-placed in $$P$$ relative to $$G$$ if the following holds.

Consider the following sequence:

$$W_1(H) = H, P_1(H) = N_P(H), N_1(H) = N_G(H)$$,

and the recursive definition is:

$$W_{i+1}(H) = W(P_i(H)), P_{i+1}(H) = N_P(W_{i+1}(H)), N_{i+1}(H) = N_G(W_{i+1}(H))$$.

Then, we require that each $$P_i(H)$$ is a $$p$$-Sylow subgroup of $$N_i(H)$$. In other words, we require that each $$W_i(H)$$ is a subgroup whose normalizer in the Sylow is Sylow in the normalizer.

For a subgroup that is not well-placed, its height is defined as the smallest $$d$$ for which $$P_d(H)$$ is not $$p$$-Sylow in $$N_d(H)$$.

Weaker properties

 * Stronger than::Subgroup whose normalizer in the Sylow is Sylow in the normalizer

Facts

 * Every p-subgroup is well-placed in some Sylow subgroup
 * Normal subgroup of Sylow subgroup is well-placed