# Extensible automorphism

In mathematics, an automorphism of a structure is said to be **extensible** if, for any embedding of that structure inside another structure, the automorphism can be lifted to the bigger structure. In group theory, an extensible automorphism of a group is an automorphism that can be lifted to an automorphism of any group in which it is embedded. A group automorphism is extensible if and only if it is an inner automorphism, Template:Harv.

A times extensible automorphism of a group is defined inductively as an automorphism that can be lifted to a times extensible automorphism for any embedding, where a 0 times extensible automorphism is simply any automorphism. An automorphism that is times extensible for all is termed an extensible automorphism. The extensible automorphisms of a group form a subgroup for every .

Here are some properties in increasing order of generality:

- The only extensible automorphism of an abelian group (extensible to arbitrary groups, not just to abelian groups) is the identity map.
- Every extensible automorphism of a finite group is an IA automorphism, that is, it acts as identity on the Abelianization.
- If a group has a homomorphic image acting on another group such that the other group is characteristic in the semidirect product and the homomorphic image is a central factor in its normalizer in the semidirect product then any extensible automorphism of the group must get quotiented to an inner automorphism of its homomorphic image.

## References

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