Locally operator

Symbol-free definition
The locally operator is a map from the group property space to itself that inputs a group property $$p$$ and outputs the property of being a group in which every finitely generated subgroup satisfies property $$p$$.

In the case where $$p$$ is a subgroup-closed group property, this is equivalent to saying that the group is isomorphic to a direct limit of groups with property $$p$$.

Definition with symbols
The locally operator is a map from the group property space to itself that inputs a group property $$p$$ and outputs the property of being a group $$G$$, such that every finitely generated subgroup of $$G$$ satisfies property $$p$$ as an abstract group.

In the case where $$p$$ is a subgroup-closed group property, this is equivalent to saying that the group is isomorphic to a direct limit of groups with property $$p$$.

A group property is termed local if it is a fixed-point under the locally operator.

Application
Important instances of application of the locally operator: