Monomial automorphism: Difference between revisions
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{{further|[[monomial map]]}} | {{further|[[monomial map]]}} | ||
===Definition with symbols=== | |||
An automorphism <math>f:G \to G</math> is termed a '''monomial automorphism''' if there exists a [[word]] <math>w(x,y_1,y_2,\ldots,y_n)</math> and fixed elements <math>a_1,a_2,\ldots,a_n \in G</math> such that for any <math>g \in G</math>: | |||
<math>f(g) = w(g,a_1,a_2,\ldots,a_n)</math> | |||
If we remove the condition of <math>f</math> being an automorphism, we get the more general notion of a [[monomial map]]. | |||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Inner automorphism]]: {{proofat|[[Inner implies monomial]]}} | |||
* [[Universal power automorphism]] | |||
* [[Strongly monomial automorphism]] | |||
===Weaker properties=== | |||
* [[Monomially generated automorphism]] | |||
* [[Intrinsically continuous automorphism]] | |||
==Metaproperties== | |||
{{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 | |||
Revision as of 13:48, 1 April 2007
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]
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.
Relation with other properties
Stronger properties
- Inner automorphism: For full proof, refer: Inner implies monomial
- Universal power automorphism
- Strongly monomial automorphism
Weaker properties
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