# 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 a 'group in which every automorph-conjugate subgroup is characteristic if it satisfies the following equivalent conditions:

## 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 (automorph-conjugate subgroup) satisfies the second property (characteristic subgroup), and vice versa.
View other group properties obtained in this way

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

## Metaproperties

Metaproperty Satisfied? Proof Statement with symbols
subgroup-closed group property No ACIC is not subgroup-closed It is possible to have a group $G$ and a subgroup $H$ of $G$ such that $G$ is ACIC but $H$ is not ACIC.
quotient-closed group property No ACIC is not quotient-closed It is possible to have a group $G$ and a normal subgroup $H$ of $G$ such that $G$ is ACIC but the quotient group $G/H$ is not ACIC.
normal subgroup-closed group property No ACIC is not normal subgroup-closed It is possible to have a group $G$ and a normal subgroup $H$ of $G$ such that $G$ is ACIC but $H$ is not ACIC.
finite direct product-closed group property No ACIC is not finite direct product-closed It is possible to have groups $G_1$ and $G_2$ such that both $G_1$ and $G_2$ are ACIC but the external direct product $G_1 \times G_2$ is not ACIC.
characteristic subgroup-closed group property Yes ACIC is characteristic subgroup-closed Suppose $G$ is an ACIC-group and $H$ is a characteristic subgroup of $G$. Then, $H$ is also an ACIC-group.
characteristic quotient-closed group property Yes ACIC is characteristic quotient-closed Suppose $G$ is an ACIC-group and $H$ is a characteristic subgroup of $G$. Note that, since characteristic implies normal, it makes sense to talk of the quotient group $G/H$. This quotient group must also be ACIC.

## 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
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.