Meta operator: Difference between revisions

Revision as of 09:24, 25 May 2007

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 $N ◃ G$ 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

@@ Line 10: / Line 10: @@
 The '''meta operator''' is a map from the [[group property space]] to itself defined as follows: it takes as input a [[group property]] <math>p</math> and outputs the group property <math>q</math> defined as follows:
+A group <math>G</math> has property <math>q</math> if there is a normal subgroup <math>N \triangleleft G</math> such that <math>N</math> and <math>G/N</math> both have property <math>p</math> (as abstract groups).
 ==Application==