Monomial map

This article defines a function property, viz a property of functions from a group to itself

Definition

Definition with symbols

A function $f:G\to G$ is termed a monomial map if there exists a word $w(x,y_{1},y_{2},\ldots ,y_{n})$ and fixed elements $a_{1},a_{2},\ldots ,a_{n}\in G$ such that for any $g\in G$:

$f(g)=w(g,a_{1},a_{2},\ldots ,a_{n})$

A monomial map which defines an endomorphism is termed a monomial endomorphism. A monomial map which defines an automorphism is termed a monomial automorphism.

Relation with other properties

Inner automorphisms

An inner automorphism is a map of the form $g\mapsto aga^{-1}$ where $a\in G$ is a fixed element. It thus fits into the definition of a monomial map. In fact, inner automorphisms are monomial automorphisms.

Universal power maps

A universal power map is a map of the form $x\mapsto x^{n}$ where $n$ is a fixed integer. clearly a universal power map is a monomial map, in fact, it is a monomial map with no parameters.

In particular universal power automorphisms are monomial automorphisms and universal power endomorphisms are monomial endomorphisms.