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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The version of this for finite groups is at: finite ACIC-group

Definition

A group is termed a 'group in which every automorph-conjugate subgroup is 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 (automorph-conjugate subgroup) satisfies the second property (characteristic subgroup), 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
• Frattini-embedded normal-realizable group: For full proof, refer: Frattini-embedded normal-realizable implies ACIC
• Dedekind group
• Hereditarily ACIC-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
• ACIC-embeddable group

Metaproperties

Metaproperty	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	ACIC is not subgroup-closed	It is possible to have a group $G$ and a subgroup $H$ of $G$ such that $G$ is ACIC but $H$ is not ACIC.
quotient-closed group property	No	ACIC is not quotient-closed	It is possible to have a group $G$ and a normal subgroup $H$ of $G$ such that $G$ is ACIC but the quotient group $G/H$ is not ACIC.
normal subgroup-closed group property	No	ACIC is not normal subgroup-closed	It is possible to have a group $G$ and a normal subgroup $H$ of $G$ such that $G$ is ACIC but $H$ is not ACIC.
finite direct product-closed group property	No	ACIC is not finite direct product-closed	It is possible to have groups $G_1$ and $G_2$ such that both $G_1$ and $G_2$ are ACIC but the external direct product $G_1 \times G_2$ is not ACIC.
characteristic subgroup-closed group property	Yes	ACIC is characteristic subgroup-closed	Suppose $G$ is an ACIC-group and $H$ is a characteristic subgroup of $G$. Then, $H$ is also an ACIC-group.
characteristic quotient-closed group property	Yes	ACIC is characteristic quotient-closed	Suppose $G$ is an ACIC-group and $H$ is a characteristic subgroup of $G$. Note that, since characteristic implies normal, it makes sense to talk of the quotient group $G/H$. This quotient group must also be ACIC.

Testing

GAP code

One can write code to test this group property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View other GAP-codable group properties | View group properties with in-built commands

The following GAP code can be used to check whether a group is ACIC:

AutomorphicImage := function(a,K)
		 local L, g;
		 L := List([]);
		 for g in Set(K) do
		     Add(L,g^a);
		 od;
		 return Group(L);
end;;

IsAutomorphConjugateSubgroup := function(G,H)
		     local A, s;
		     A := AutomorphismGroup(G);
		     for s in A do
		     	 if not (AutomorphicImage(s,H) in ConjugateSubgroups(G,H)) then
			    return false;
			    fi;
			 od;
		     return true;
end;;

IsACIC := function(G)
       local H;
       if IsAbelian(G) then return true; fi;
       for H in List(ConjugacyClassesSubgroups(G),Representative) do
       	   if IsAutomorphConjugateSubgroup(G,H) and not IsNormal(G,H) then return false; fi;
od;
	return true;
end;;

To do the test, enter:

IsACIC(G)

where $G$ is the group that needs to be tested. The code works only for finite groups, and as such, is extremely inefficient.