Abelian-extensible automorphism
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BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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
View other terms related to Extensible automorphisms problem | View facts related to Extensible automorphisms problem
Definition
Symbol-free definition
An automorphism of an abelian group is termed abelian-extensible if it can be extended to an automorphism for any embedding of the group in an abelian group.
Definition with symbols
An automorphism σ of an Abelian group G is termed abelian-extensible if, for any embedding of G as a subgroup of an abelian group H, there exists an automorphism σ' of H whose restriction to G equals σ.
This is equivalent to the following: if D is the unique largest divisible abelian subgroup of G, then σ induces a power automorphism on the quotient group G / D.
Equivalence of definitions
For full proof, refer: Classification of abelian-extensible automorphisms
Relation with other properties
Facts
- Divisible abelian group implies every automorphism is abelian-extensible
- Abelian-extensible automorphism not implies power map
- Subgroup of abelian group not implies abelian-extensible automorphism-invariant
Related facts
- Divisible abelian group implies every endomorphism is abelian-extensible
- Free abelian group implies every automorphism is abelian-quotient-pullbackable
- Free abelian group implies every endomorphism is abelian-quotient-pullbackable
