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 $\sigma$ 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).

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:
  • Inner automorphism
  • Subnormal-extensible automorphism
• Iteratively normal-extensible automorphism viz $\alpha$-normal extensible automorphism for an ordinal $\alpha$ that is at least 1
• Normal series-extensible automorphism

Weaker properties

• Characteristic-extensible automorphism
• Central factor-extensible automorphism

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