Extensible automorphism-invariant equals normal

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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If the ambient group is a finite group, this property is equivalent to the property: normality
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Definition

Symbol-free definition

A subgroup of a group is termed extensible automorphism-invariant if any extensible automorphism on the whole group takes the subgroup to within itself.

Here an extensible automorphism means an automorphism of the group that can be extended to an automorphism for any group containing it.

Definition with symbols

A subgroup $H$ of a group $G$ is termed extensible automorphism-invariant if for any extensible automorphism $\sigma$ of $G$, $\sigma (H)\subseteq H$.

Formalisms

The subgroup property of being extensible automorphism-invariant can be expressed using the function restriction formalism in the following ways:

• As the invariance property with respect to the function property of being an extensible automorphism, viz:

Extensible automorphism $\to$ Function

• As the function restriction formal expression with the left side being extensible automorphisms and the right side being automorphisms, viz:

Extensible automorphism $\to$ Automorphism

Further, the latter expression is right tight, viz we cannot replace automorphism by anything stronger. This follows by a similar reasoning to that for right tightness for the corresponding expression of normality.

Relation with other properties

Stronger properties

• Strongly potentially characteristic subgroup
• Potentially characteristic subgroup
• Potentially relatively characteristic subgroup
• Pushforwardable automorphism-invariant subgroup

Weaker properties

• Normal subgroup

Conjecture of equalling normality

This property is conjectured to equal the property: normality

The conjecture has already been settled for groups that are finite or that haev some finiteness conditions. Note that this is equivalent to the conjecture that every extensible automorphism is quotientable.