# Monomial automorphism

From Groupprops

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)

View other automorphism properties OR View other function properties

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 is termed a **monomial automorphism** if there exists a word and fixed elements such that for any :

If we remove the condition of 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

View other such automorphism properties

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.

## Relation with other properties

### Stronger properties

- Inner automorphism:
`For full proof, refer: Inner implies monomial` - Universal power automorphism
- Strong monomial automorphism

### Weaker properties

- Monomially generated automorphism
- Intrinsically continuous automorphism
- Weakly normal automorphism:
`For full proof, refer: Monomial implies weakly normal`

## 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