Maximal Sylow intersection

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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

Let $G$ be a finite group and $H$ be a subgroup of $G$ . Let $p$ be a prime number. We say that $H$ is a maximal $p$ -Sylow intersection in $G$ if the following two conditions hold:

There exist two distinct $p$ -Sylow subgroups $P,Q$ of $G$ such that $H=P\cap Q$ .
For any two distinct $p$ -Sylow subgroups $R,S$ of $G$ , the order of $R\cap S$ is not more than the order of $H$ .