Monomial automorphism

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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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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
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

Weaker properties

Metaproperties

Template:Monoid-closed ap

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