# Monomial automorphism

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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## Contents

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
This is a variation of inner automorphism|Find other variations of inner automorphism |

## Definition

A monomial automorphism is a monomial map that is also an automorphism

Further information: monomial map

### Definition with symbols

An automorphism $f:G \to G$ is termed a monomial automorphism 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 $g \in G$: $f(g) = w(g,a_1,a_2,\ldots,a_n)$

If we remove the condition of $f$ being an automorphism, we get the more general notion of a monomial map.

## Formalisms

### Variety formalism

This automorphism property can be described in the language of universal algebra, viewing groups as a variety of algebras
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Viewing the variety of groups as a variety of algebras, monomial automorphisms are precisely the formula automorphisms of this variety. This is direct from the definition.

## Metaproperties

A product of monomial automorphisms is a monomial automorphism. This follows from the following two facts:

• A product of automorphisms is an automorphism
• A product of monomial maps is a monomial map