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)
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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 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