# 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:

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

## 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
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;;```