Large operator

Definition
Given a subgroup property $$p$$, the large operator applied to the property $$p$$ gives the following subgroup property $$q$$. A subgroup $$H$$ of $$G$$ satisfes property $$q$$ in $$G$$ if given any subgroup $$K$$ satisfying $$p$$ in $$G$$:

$$H \cap K$$ is trivial $$\implies K$$ is trivial

Application
Some important instances of application of the large operator:



Properties
Note that the property of being large with respect to $$p$$ says something like: for every subgroup with property $$p$$. Thus, the more subgroups there are with property $$p$$, the harder it is to be $$p$$-large. More formally if $$p \le q$$, then $$q$$-large $$\implies p$$-large.