# Meta operator

From Groupprops

*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

### Property-theoretic definition

The **meta operator** is a map from the group property space to itself, that takes as input a group property and outputs the *square* of under the group extension operator.

### Definition with symbols

The **meta operator** is a map from the group property space to itself defined as follows: it takes as input a group property and outputs the group property defined as follows:

A group has property if there is a normal subgroup such that and both have property (as abstract groups).

## Application

Important instances of application of the meta operator:

- Metacyclic group: obtained from Cyclic group
- Metabelian group: obtained from Abelian group
- Metanilpotent group: obtained from nilpotent group