Diagonal-in-square operator

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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.