Finite-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 Finite-extensible endomorphism, all facts related to Finite-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

This article defines a function property, viz a property of functions from a group to itself

Definition

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

It turns out that any finite-extensible endomorphism must be an automorphism.

Facts

• Finite-extensible endomorphism implies trivial or automorphism: This in turn follows from the fact that every finite group is a subgroup of a finite simple group.