Abelian-extensible automorphism

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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
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 $\sigma$ 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 ${\sigma }^{\prime }$ of $H$ whose restriction to $G$ equals $\sigma$.

Relation with other properties

Other related properties

• Abelian-extensible endomorphism

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