Simple-complete subgroup property

Symbol-free definition
A subgroup property $$p$$ is said to be simple-complete if it satisfies the following:


 * $$p$$ is trim, viz in every group, the trivial subgroup and the whole group satisfy property $$p$$
 * Every group can be embedded in a group that is $$p$$-simple (that is, that satisfies the group property obtained by applying the simple group operator to $$p$$). In other words, every group can be embedded into a group that has no proper nontrivial subgroup satisfying $$p$$

Related properties

 * Finite-simple-complete subgroup property: This is a trim subgroup property where every finite group can be embedded in a finite group which is simple with respect to that property