Subgroup intersection restriction expression

Definition
A subgroup intersection restriction expression is an expression of a subgroup property in terms of two subgroup properties. The subgroup intersection restriction expression corresponding to subgroup properties $$p$$ and $$q$$ is denoted as:

$$p \to q$$

and is defined as follows: a subgroup $$H \le G$$ satisfies proeprty $$p \to q$$ if for any subgroup $$K$$ satisfying property $$p$$ in $$G$$, $$H \cap K$$ satisfies $$q$$ in $$H$$.