Diagonal-in-square operator

Definition
The diagonal-in-square operator is an operator that takes as input a subgroup property and outputs a group property as follows.

Suppose $$p$$ is a subgroup property. The diagonal-in-square operator on $$p$$ gives the property of being a group $$G$$ such that, in the external direct product $$G \times G$$, the diagonal subgroup $$\{ (g,g) \mid g \in G \}$$ satisfies property $$p$$.

Examples

 * Applying the diagonal-in-square operator to the property of being a characteristic subgroup gives the property of being thetrivial group.
 * Applying the diagonal-in-square operator to the property of being a normal subgroup gives the property of being an abelian group.
 * Applying the diagonal-in-square operator to the property of being a subnormal subgroup gives the property of being a nilpotent group.