# Own-variety-extensible automorphism

## Contents

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This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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## Definition

An automorphism $\sigma$ of a group $G$ is termed an own-variety-extensible automorphism if, for any group $H$ containing $G$ such that $H$ is in the subvariety generated by $G$ in the variety of groups, $\sigma$ extends to an automorphism of $H$.

In other words, an own-variety-extensible automorphism is an automorphism of a group that satisfies the following equivalent conditions:

• It is variety-extensible for some subvariety of the variety of groups.
• It is variety-extensible for the subvariety of the variety of groups generated by that particular group.