Finite-extensible endomorphism

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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 ${\alpha }^{\prime }$ of $H$ such that the restriction of ${\alpha }^{\prime }$ 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.