Simple group operator

Symbol-free definition
The simple group operator is an operator from the collection of trim subgroup properties to the group property space. It acts as follows: given a trim subgroup property $$p$$, a group is termed $$p$$-simple if it is nontrivial and has no proper nontrivial subgroup satisfying $$p$$.

(A subgroup property is trim if it is satisfied by both the whole group and the trivial subgroup).

Note that the trivial group is not considered $$p$$-simple.

Examples

 * Normal subgroup gives rise to simple group
 * Characteristic subgroup gives rise to characteristically simple group
 * Direct factor gives rise to directly indecomposable group
 * Retract gives rise to semidirectly indecomposable group
 * Ascendant subgroup gives rise to strictly simple group
 * Serial subgroup gives rise to absolutely simple group

Simple-complete property
A trim property $$p$$ is termed simple-complete if every group can be embedded as a proper subgroup of a $$p$$-simple group.

Core operator
Suppose the given property $$p$$ is trim and join-closed. Then, in a $$p$$-simple group, the $$p$$-core of any proper subgroup is trivial. That is, every proper subgroup is $$p$$-core-free.

If $$p$$ is a simple-complete property, then every subgroup of a group is potentially core-free with respect to $$p$$.

Closure operator
Suppose $$p$$ is intersection-closed and trim. Then, in a $$p$$-simple group, the $$p$$-closure of any nontrivial subgroup is the whole group. Equivalently, every nontrivial group is contra-$$p$$.

Thus, if $$p$$ is a simple-complete property, then every nontrivial subgroup of a group is 'potentially contra$$p$$.