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 $p$ 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:

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