# Locally operator

*This article defines a group property modifier (a unary group property operator) -- viz an operator that takes as input a group property and outputs a group property*

*This property modifier is idempotent and a property is a fixed-point, or equivalently, an image of this if and only if it is a:*local group property

## Definition

### Symbol-free definition

The **locally operator** is a map from the group property space to itself that inputs a group property and outputs the property of being a group in which every finitely generated subgroup satisfies property .

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

### Definition with symbols

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

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

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:

- locally cyclic group: obtained from cyclic group
- locally finite group: obtained from finite group
- locally nilpotent group: obtained from nilpotent group