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 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
• Dedekind 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

Metaproperties

Subgroups

This group property is not subgroup-closed, viz., we can have a group satisfying the property, with a subgroup not satisfying the property

Not every subgroup of an ACIC-group is ACIC. For instance, the non-Abelian group of exponent $p^{2}$ and order $p^{3}$ is not ACIC, but its direct product with the cyclic group of order $p$ is ACIC.

Quotients

This group property is not quotient-closed, viz., we could have a group with the property and a quotient group of that group that does not have the property

The same example as used above works (observe that a direct factor is both a subgroup, and a quotient).

Direct products

This group property is not direct product-closed. A direct product of groups with this property, need not have this property.

A direct product of ACIC-groups need not be ACIC. In fact, even a direct product of an ACIC group and an Abelian group need not be ACIC. Specifically the direct product of the dihedral group of order eight and the cyclic group of order four, is not ACIC, though both groups are.

Characteristic subgroups

This group property is characteristic subgroup-closed: any characteristic subgroup of a group with the property, also has the property
View characteristic subgroup-closed group properties]]

Any characteristic subgroup of an ACIC-group is ACIC. Equivalently, any automorph-conjugate subgroup of an ACIC-group is ACIC. For full proof, refer: ACIC is characteristic subgroup-closed

Characteristic quotients

This group property is characteristic quotient-closed: the quotient group by any characteristic subgroup, of a group with this property, also has this property
View other characteristic quotient-closed group properties

The quotient group of an ACIC-group by a characteristic subgroup is again ACIC. For full proof, refer: ACIC is characteristic quotient-closed

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 no 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.