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

Automorph-conjugate subgroup = Characteristic subgroup

Stronger properties

 * Abelian group
 * Finite-Frattini-realizable group
 * Frattini-embedded normal-realizable group:
 * Dedekind group
 * Hereditarily ACIC-group

Weaker properties

 * Nilpotent group (for finite groups): This follows from the fact that Sylow subgroups are automorph-conjugate. The implication does not hold for infinite groups. . Also, the converse implication does not hold even for finite groups.
 * ACIC-embeddable group

Testing
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.