# Variety-extensible automorphism

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This article defines a property that can be evaluated for an automorphism of an algebra in a variety of algebras. The evaluation of that property depends on the ambient variety, and not just on the automorphism or the algebra.

View all such properties

## Definition

Let be a variety of algebras, and be an algebra in . An automorphism of is termed *extensible'* over the variety if, whenever is embedded as a subalgebra of an algebra of , there exists an automorphism of such that the restriction of to is .

## Particular cases

### Variety of sets

In the variety of sets, *every* automorphism is extensible. In other words, given any set and a subset, a permutation of the subset always extends to a permutation of the whole set.

The idea here is that adding more elements to a set does not destroy the inherent symmetry between the elements already there.

### Variety of groups

`Further information: extensible automorphism`

In the variety of groups, every inner automorphism is extensible. The extensible automorphisms conjecture states that every extensible automorphism is inner.