Levi operator

Symbol-free definition
The Levi operator is a map from the group property space to the group property space that takes as input a group property $$p$$ and outputs the property of being a group where every point-closure (viz the normal closure of every element) satisfies the group property $$p$$ as an abstract group.

Definition with symbols
The Levi operator is a map from the group property space to the group property space that takes as input a group property $$p$$ and gives as output the property of being a group $$G$$ such that for any element $$x$$ in $$G$$, the normal closure of $$x$$ satisfies property $$p$$ as an abstract group.

Application
Important instances of application of the Levi operator: