Meta 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

Property-theoretic definition

The meta operator is a map from the group property space to itself, that takes as input a group property $p$ and outputs the square of $p$ 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 $p$ and outputs the group property $q$ defined as follows:

A group $G$ has property $q$ if there is a normal subgroup $N\triangleleft G$ such that $N$ and $G/N$ both have property $p$ (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