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.