Transfer condition

Definition with symbols
A subgroup property $$p$$ is said to satisfy the transfer condition if whenever $$H$$ satisfies property $$p$$ as a subgroup of $$G$$, and $$K$$ is a subgroup of $$G$$, then $$H$$ &cap; $$K$$ satisfies property $$p$$ as a subgroup of $$K$$.

Formalisms
Consider the procedure $$P$$ that takes as input a group-subgroup pair $$H \le G$$, and outputs all group-subgroup pairs $$H \cap K \le K$$ for $$K \le G$$. The transfer condition is the single-input-expressible metaproperty corresponding to procedure $$P$$: a subgroup property $$p$$ satisfies the transfer condition if $$H \le G$$ satisfying property $$p$$ implies that all pairs $$H \cap K \le K$$ also satisfy property $$p$$.

Stronger metaproperties

 * Weaker than::Inverse image condition
 * Weaker than::Strongly UL-intersection-closed subgroup property

Weaker metaproperties

 * Stronger than::Intermediate subgroup condition

Conjunction implications

 * Any transitive subgroup property that satisfies the transfer condition is also finite-intersection-closed.