Extensible endomorphism

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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
VIEW: Definitions built on this | Facts about this: (facts closely related to Extensible endomorphism, all facts related to Extensible endomorphism) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

Definition

Let $G$ be a group and $\alpha$ be an endomorphism of $G$. We say that $\alpha$ is extensible if, for any group $H$ containing $G$, there is an endomorphism $\alpha '$ of $H$ such that the restriction of $\alpha '$ to $G$ equals $\alpha$.

Related terms

An extensible automorphism is an automorphism that can always be extended to an automorphism for any bigger group. Every extensible automorphism is an extensible endomorphism and also an automorphism. However, there is no a priori reason that every extensible endomorphism that is also an automorphism should be an extensible automorphism.

A finite-extensible endomorphism is an endomorphism of a finite group that can always be extended to an endomorphism for any finite group containing it.

Facts

• Finite-extensible endomorphism implies trivial or automorphism