# Normal-extensible automorphism

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

## Definition

### Symbol-free definition

An automorphism of a group is termed normal-extensible if, for any embedding of the group as a normal subgroup of another group, the automorphism can be extended to an automorphism of the bigger group.

### Definition with symbols

An automorphism $\sigma$ of a group $G$ is termed normal-extensible if, for any embedding of $G$ as a normal subgroup of another group $H$ there is an automorphism $\sigma'$ of $H$ such that the restriction of $\sigma'$ to $G$ is $\sigma$.

## Formalisms

### In terms of the qualified extensibility operator

This property is obtained by applying the qualified extensibility operator to the property: normality
View other properties obtained by applying the qualified extensibility operator

The property of normal-extensibility arises by applying the qualified extensibility operator with the qualifying property being normality and the the automorphism property being the tautology (that is, the property of being any automorphism).

## Facts

More facts about characterizing normal-extensible automorphisms can be found at the normal-extensible automorphisms problem page.

## Metaproperties

### Group-closedness

This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
View a complete list of group-closed automorphism properties

The collection of normal-extensible automorphisms of a group form a subgroup of the automorphism group. This follows from the general fact that the qualified extensibility operator is a group-closure-preserving automorphism property operator.

For full proof, refer: qualified extensibility operator is group-closure-preserving