# Levi 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*

## Definition

### 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 and outputs the property of being a group where every point-closure (viz the normal closure of every element) satisfies the group property 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 and gives as output the property of being a group such that for any element in , the normal closure of satisfies property as an abstract group.

## Application

Important instances of application of the Levi operator:

- Dedekind group: obtained from cyclic group
- 2-Engel group: obtained from Abelian group
- Fitting group: obtained from nilpotent group