Simple non-abelian group

Symbol-free definition
A group is said to be a simple non-Abelian group if:


 * It is simple viz it has no proper nontrivial subgroups
 * It is not Abelian viz it is not true that any two elements in the group commute.

Every subgroup-defining function gives trivial group or whole group
Since any subgroup-defining function (such as the center, the commutator subgroup, the Frattini subgroup, the Fitting subgroup etc.) returns a characteristic subgroup of the whole group, and since every characteristic subgroup is normal, any subgroup obtained via a subgroup-defining function must be either trivial or the whole group. This, combined with the fact that the group is non-Abelian, tells us the following:


 * The center must be trivial -- in other words, the group is centerless
 * The commutator subgroup must be the whole group -- in other words, the group is perfect