# Finite-extensible endomorphism

## Contents

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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 $\alpha'$ of $H$ such that the restriction of $\alpha'$ to $G$ equals $\alpha$.

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