# Finite-extensible endomorphism

From Groupprops

## Contents |

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

## Definition

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

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.