Simple group operator
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 , a group is termed -simple if it is nontrivial and has no proper nontrivial subgroup satisfying .
(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 -simple.
- 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
A trim property is termed simple-complete if every group can be embedded as a proper subgroup of a -simple group.
Suppose the given property is trim and join-closed. Then, in a -simple group, the -core of any proper subgroup is trivial. That is, every proper subgroup is -core-free.
If is a simple-complete property, then every subgroup of a group is potentially core-free with respect to .
Suppose is intersection-closed and trim. Then, in a -simple group, the -closure of any nontrivial subgroup is the whole group. Equivalently, every nontrivial group is contra-.
Thus, if is a simple-complete property, then every nontrivial subgroup of a group is 'potentially contra.