# Finitarily extensible automorphism

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)
View other automorphism properties OR View other function properties
This is a variation of extensible automorphism|Find other variations of extensible automorphism |

## Definition

### Symbol-free definition

An automorphism of a group is said to be finitarily extensible if it can be extended to an automorphism of any group containing it, in which it has finite index.

When the starting group is finite, this reduces to the notion of finite-extensible automorphism.

### Definition with symbols

An automorphism $\sigma$ of a group $G$ is said to be finitarily extensible if for any $H$ containing $G$ such that $[H:G]$ is finite, there exists an automorphism $\sigma'$ of $H$ such that the restriction of $\sigma'$ to $G$ is $\sigma$.