Monomial automorphism

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

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
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.

Stronger properties

 * Weaker than::Inner automorphism:
 * Weaker than::Universal power automorphism
 * Weaker than::Strong monomial automorphism

Weaker properties

 * Stronger than::Monomially generated automorphism
 * Stronger than::Intrinsically continuous automorphism
 * Stronger than::Weakly normal automorphism:

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