# Diagonal-in-square operator

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## Definition

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

Suppose is a subgroup property. The diagonal-in-square operator on gives the property of being a group such that, in the external direct product , the diagonal subgroup satisfies property .

## 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.