Group in which every automorph-conjugate subgroup is characteristic

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group is termed ACIC or automorph-conjugate implies characteristic if it satisfies the following equivalent conditions:

• Every automorph-conjugate subgroup of it is characteristic
• Every automorph-conjugate subgroup is normal.

Formalisms

In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property ({{{1}}}Property "Defining ingredient" (as page type) with input value "{{{1}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.) satisfies the second property ({{{2}}}Property "Defining ingredient" (as page type) with input value "{{{2}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.), and vice versa.
View other group properties obtained in this way

The property of being an ACIC-group can be viewed as the collapse:

Automorph-conjugate subgroup = Characteristic subgroup

Relation with other properties

Stronger properties

• Abelian group
• Finite-Frattini-realizable group

Weaker properties

• Nilpotent group (for finite groups): This follows from the fact that Sylow subgroups are automorph-conjugate. For full proof, refer: ACIC implies nilpotent (finite groups)]]The implication does not hold for infinite groups. For full proof, refer: ACIC not implies nilpotent (infinite groups). Also, the converse implication does not hold even for finite groups. For full proof, refer: Nilpotent not implies ACIC