Monomial map

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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.