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 $w(x,y_1,y_2,\dots ,y_n)$ and fixed elements $a_1,a_2,\dots ,a_n\in G$ such that for any $g\in G$:

$f(g)=w(g,a_1,a_2,\dots ,a_n)$

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

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
• Normal automorphism: For full proof, refer: Monomial implies 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