# 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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## Contents

## 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

## Distribution

To classify and understand finite ACIC-groups, it suffices to classify and understand the -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 inGAP (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;;