# Group in which every automorph-conjugate subgroup is characteristic

From Groupprops

(Redirected from ACIC-group)

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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:

- Every automorph-conjugate subgroup of it is characteristic
- Every automorph-conjugate subgroup is normal.

## 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:

Automorph-conjugate subgroup = Characteristic subgroup

## Relation with other properties

### Stronger properties

- Abelian group
- Finite-Frattini-realizable group
- Frattini-embedded normal-realizable group:
`For full proof, refer: Frattini-embedded normal-realizable implies ACIC` - Dedekind group
- Hereditarily ACIC-group

### Weaker properties

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

## Metaproperties

Metaproperty | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | No | ACIC is not subgroup-closed | It is possible to have a group and a subgroup of such that is ACIC but is not ACIC. |

quotient-closed group property | No | ACIC is not quotient-closed | It is possible to have a group and a normal subgroup of such that is ACIC but the quotient group is not ACIC. |

normal subgroup-closed group property | No | ACIC is not normal subgroup-closed | It is possible to have a group and a normal subgroup of such that is ACIC but is not ACIC. |

finite direct product-closed group property | No | ACIC is not finite direct product-closed | It is possible to have groups and such that both and are ACIC but the external direct product is not ACIC. |

characteristic subgroup-closed group property | Yes | ACIC is characteristic subgroup-closed | Suppose is an ACIC-group and is a characteristic subgroup of . Then, is also an ACIC-group. |

characteristic quotient-closed group property | Yes | ACIC is characteristic quotient-closed | Suppose is an ACIC-group and is a characteristic subgroup of . Note that, since characteristic implies normal, it makes sense to talk of the quotient group . This quotient group must also be ACIC. |

## 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.

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 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 is the group that needs to be tested. The code works only for finite groups, and as such, is extremely inefficient.