# Normal-extensible automorphism

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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 of a group is termed **normal-extensible** if, for any embedding of as a normal subgroup of another group there is an automorphism of such that the restriction of to is .

## 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).

## Relation with other properties

### Stronger properties

- Extensible automorphism:
*For proof of the implication, refer extensible implies normal-extensible and for proof of its strictness (i.e. the reverse implication being false) refer normal-extensible not implies extensible*.. Also related: - Iteratively normal-extensible automorphism viz -normal extensible automorphism for an ordinal that is at least 1
- Normal series-extensible automorphism

### Weaker properties

### Incomparable properties

- Normal automorphism:
`For full proof, refer: Normal not implies normal-extensible, normal-extensible not implies normal`

### Related group properties

- Group in which every automorphism is normal-extensible
- Group in which every normal-extensible automorphism is inner

## Facts

- Centerless and maximal in automorphism group implies every automorphism is normal-extensible
- Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
- NRPC conjecture implies normal-extensible equals characteristic-extensible

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`