# Single-input-expressible subgroup metaproperty

A subgroup metaproperty $\alpha$ is termed single-input-expressible if it admits the following description. Consider a procedure $P$ that takes as input a pair of a group and a subgroup, and outputs a collection of group-subgroup pairs.
A subgroup property $p$ satisfies $\alpha$ if and only if, whenever the starting pair satisfies property $p$, all the pairs in the resulting collection also satisfy property $p$.