# Own-variety-extensible automorphism

From Groupprops

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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 of a group is termed an **own-variety-extensible automorphism** if, for any group containing such that is in the subvariety generated by in the variety of groups, extends to an automorphism of .

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.