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 ${\sigma }^{\prime }$ of $H$ such that the restriction of ${\sigma }^{\prime }$ 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

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

Weaker properties

• Characteristic-extensible automorphism
• Central factor-extensible automorphism

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