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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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.

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