Monomial automorphism: Difference between revisions
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==Definition== | ==Definition== | ||
A '''monomial automorphism''' is a [[monomial map]] that is also an [[automorphism]] | A '''monomial automorphism''' is a [[monomial map]] that is also an [[automorphism]]. | ||
{{further|[[monomial map]]}} | {{further|[[monomial map]]}} | ||
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If we remove the condition of <math>f</math> being an automorphism, we get the more general notion of a [[monomial map]]. | If we remove the condition of <math>f</math> being an automorphism, we get the more general notion of a [[monomial map]]. | ||
==Formalisms== | |||
{{variety-expressible automorphism property}} | |||
Viewing the [[variety of groups]] as a [[variety of algebras]], monomial automorphisms are precisely the [[formula automorphism]]s of this variety. This is direct from the definition. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Weaker than::Inner automorphism]]: {{proofat|[[Inner implies monomial]]}} | * [[Weaker than::Inner automorphism]]: {{proofat|[[Inner implies monomial]]}} | ||
* [[Weaker than::Universal power automorphism]] | * [[Weaker than::Universal power automorphism]] | ||
* [[Weaker than:: | * [[Weaker than::Strong monomial automorphism]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
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* [[Stronger than::Monomially generated automorphism]] | * [[Stronger than::Monomially generated automorphism]] | ||
* [[Stronger than::Intrinsically continuous automorphism]] | * [[Stronger than::Intrinsically continuous automorphism]] | ||
* [[Stronger than::automorphism]]: {{proofat|[[Monomial implies normal]]}} | * [[Stronger than::Weakly normal automorphism]]: {{proofat|[[Monomial implies weakly normal]]}} | ||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 04:53, 2 May 2022
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)
View other automorphism properties OR View other function properties
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