Transfer condition operator

Definition
Suppose $$p$$ is a subgroup property. The transfer condition operator applied to $$p$$, denoted $$T(p)$$, is defined as the following property: a subgroup $$H$$ of a group $$G$$ satisfies $$T(p)$$ if, for any subgroup $$K$$ of $$G$$, $$H \cap K$$ satisfies property $$p$$ in $$K$$.