Finite ACIC-group

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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Definition

Symbol-free definition

A finite ACIC-group is a finite group that is also an ACIC-group. The following equivalent definitions are true for a finite ACIC-group:

• It is finite and ACIC: every automorph-conjugate subgroup is characteristic
• It is finite and every automorph-conjugate subgroup is normal
• It is a finite nilpotent group, and every Sylow subgroup is an ACIC-group

Thus, the study of finite ACIC-groups reduces to the study of ACIC-groups of prime power order.

Equivalence of definitions

Further information: Equivalence of definitions of finite ACIC

Relation with other properties

Stronger properties

• Finite Abelian group
• Finite-Frattini-realizable group
• Finite Dedekind group

Distribution

To classify and understand finite ACIC-groups, it suffices to classify and understand the $p$-groups that are ACIC.

For the prime 2

We can classify finite groups of prime power order into three parts: the Abelian groups, the ACIC non-Abelian groups, and the non-ACIC groups. Here is the distribution of these:

• Order 2: 1 Abelian, 0 ACIC non-Abelian, 0 non-ACIC
• Order 4: 2 Abelian, 0 ACIC non-Abelian, 0 non-ACIC
• Order 8: 3 Abelian, 2 ACIC non-Abelian, 0 non-ACIC
• Order 16: 5 Abelian, 7 ACIC non-Abelian, 2 non-ACIC
• Order 32: 7 Abelian, 23 ACIC non-Abelian, 21 non-ACIC

Not all of the ACIC non-Abelian groups are known to occur as Frattini subgroups.

For the prime 3

• Order 3: 1 Abelian, 0 ACIC non-Abelian, 0 non-ACIC
• Order 9: 2 Abelian, 0 ACIC non-Abelian, 0 non-ACIC
• Order 27: 3 Abelian, 1 ACIC non-Abelian (the group of exponent 3), 1 non-ACIC (the group of exponent 9)
• Order 81: 5 Abelian, 6 ACIC non-Abelian, 4 non-ACIC

An identical pattern holds for the corresponding powers of the prime 5.

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.
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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;;