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Extensible automorphism
From Groupprops
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BEWARE! This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
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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)
View other automorphism properties OR View other function properties
This term is related to: Extensible automorphisms problem
See more terms related to Extensible automorphisms problem OR see facts/theorems related to Extensible automorphisms problem
This is a variation of inner automorphism
View a complete list of variations of inner automorphism OR read a survey article on varying inner automorphism
This is the property of being a variety-extensible automorphism for the following variety of algebras: groups
History
This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page
Definition
Symbol-free definition
An automorphism of a group is termed extensible if it extends to an automorphism for every embedding of the group in a bigger group.
Definition with symbols
An automorphism σ of a group G is termed extensible if, for any embedding of G in any group H, there is an automorphism φ of H such that the restriction of φ to G is σ.
Such a φ is termed an extension of σ.
Formalisms
In terms of the extensibility operator
This property is obtained by applying the extensibility operator to the property: tautology
View all properties obtained by applying the extensibility operator
The property of being an extensible automorphism is obtained by applying the extensibility operator to the tautology property (that is, the property of being any automorphism).
Relation with other properties
This property is conjectured to equal the property: inner automorphism
Stronger properties
- Inner automorphism
- Infinity-extensible automorphism
- Iteratively extensible automorphism: α-extensible automorphism for α an ordinal that is at least 1
- Chain-extensible automorphism
- Pushforwardable automorphism
- Iteratively pushforwardable automorphism
Weaker properties
- Automorphism
- Automorphism that is an extensible endomorphism
- Finitarily extensible automorphism
- Normal-extensible automorphism
- Semidirectly extensible automorphism
- Permutation-extensible automorphism: For full proof, refer: Extensible implies permutation-extensible
- Subgroup-conjugating automorphism: For full proof, refer: Extensible implies subgroup-conjugating
- Normal automorphism: For full proof, refer: Extensible implies normal
Related properties
Metaproperties
Group-closedness
This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
View a complete list of group-closed automorphism properties
The extensible automorphisms of a group form a subgroup of its automorphism group. For full proof, refer: Extensibility is group-closed
This follows from the more general fact that the extensibility operator is group-closure-preserving, in other words if p is a group-closed automorphism property, so is the result of applying the extensibility operator to p.
