# CS-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 |
This term is related to: Extensible automorphisms problem
View other terms related to Extensible automorphisms problem | View facts related to Extensible automorphisms problem

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

An automorphism of a group is termed CS-extensible if it can be extended to an inner automorphism for every embedding of the group into the automorphism group of a characteristically simple group.

### Definition with symbols

Let $G$ be a group and $\sigma$ an automorphism of $G$. We say that $\sigma$ is CS-extensible if for any embedding $G \le Aut(N)$ where $N$ is a characteristically simple group, there is an inner automorphism $\phi$ of $Aut(N)$ such that the restriction of $\phi$ to $G$ is $\sigma$.