Tame Sylow intersection

Definition
Suppose $$G$$ is a finite group and $$p$$ is a prime number dividing the order of $$G$$. Suppose, further, that $$P, Q$$ are (possibly equal) $$p$$-Sylow subgroups. The intersection $$P \cap Q$$ is termed a tame Sylow intersection if $$N_P(P \cap Q)$$ and $$N_Q(P \cap Q)$$ are both $$p$$-Sylow subgroups of $$N_G(P \cap Q)$$.

Note that this property depends on the specific choice of subgroups $$P, Q$$ we pick, and not just on their intersection. Specifically, we are interested in the embedding of $$P \cap Q$$ inside $$P$$ relative to $$G$$.

(Some definitions add a distinctness requirement from the two $$p$$-Sylow subgroups, in which case a $$p$$-Sylow subgroup ceases to be a tame intersection with itself).

Stronger properties

 * Weaker than::Well-placed tame Sylow intersection: This places the added requirement that the intersection $$P \cap Q$$ be a well-placed subgroup inside $$P$$.
 * Weaker than::Maximal Sylow intersection:

Weaker properties

 * Stronger than::Subgroup whose normalizer in the Sylow is Sylow in the normalizer: By definition, the tame Sylow intersection has this property in both $$P$$ and $$Q$$.

Textbook references

 * , Page 240, Section 7.2 (Alperin's theorem)