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.