Minimum condition operator

Definition
The minimum condition operator is an operator from the subgroup property space to the group property space defined as follows: the minimum condition operator on a subgroup property $$p$$ gives the property of being a group such that the following hold:


 * Every collection of subgroups of the group satisfying property $$p$$ has a minimal element
 * Every descending chain of subgroups, each with property $$p$$, stabilizes in finitely many steps