# Automorph-conjugate subgroup

## Contents

## Definition

### Names

There is no standard name for this property, but both **automorph-conjugate** and **intravariant** have been used.

### Equivalent definitions in tabular format

No. | Shorthand | A subgroup of a group is termed automorph-conjugate or intravariant if ... | A subgroup of a group is termed automorph-conjugate or intravariant if... (right-action convention) | A subgroup of a group is termed automorph-conjugate or intravariant if... (left-action convention) |
---|---|---|---|---|

1 | conjugate to automorphs | any automorphic subgroup (i.e. any subgroup to which it can go via an automorphism of the whole group), is also conjugate to the subgroup. | for any , there exists such that . | for every , there exists such that . |

2 | product with normalizer in normal embedding is whole group | whenever the bigger group is embedded as a normal subgroup of some ambient group, the product of the bigger group with the normalizer of the smaller group in the ambient group, is the whole group. | for any group containing as a normal subgroup, we have . | (same as right action convention statement). |

3 | conjugate to automorphs via a generating set of the automorphism group | (choose a generating set for the automorphism group) any automorphic subgroup to it via an automorphism in the generating set is conjugate to it. | (choose a generating set of ), we have that for any , there exists such that . | (choose a generating set of ), we have that every , there exists such that . |

### Equivalence of definitions

The equivalence of definitions (1) and (2) follows Frattini's argument.

For the equivalence of definitions (1) and (3):

- (1) implies (3) is clear.
- For (3) implies (1), we essentially use that the subgroup of inner automorphisms is normal in the subgroup of automorphisms.
`Further information: Automorph-conjugate iff conjugate to image under a generating set of automorphism group`

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

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

This is a variation of characteristic subgroup|Find other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup

## History

This subgroup property was studied somewhat by Wielandt, who dubbed them **intravariant subgroup**s.

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### Relation implication expression

This subgroup property is a relation implication-expressible subgroup property: it can be defined and viewed using a relation implication expression

View other relation implication-expressible subgroup properties

The subgroup property of being automorph-conjugate can be expressed as automorphic subgroups conjugate subgroups. In other words, is automorph-conjugate in iff for every automorph of , and are conjugate subgroups.

## Examples

### Extreme examples

- The trivial subgroup in any group is an automorph-conjugate subgroup.
- Every group is automorph-conjugate as a subgroup of itself.

More generally, any characteristic subgroup of a group is automorph-conjugate.

### High-occurrence examples

- In a cyclic group, every subgroup is characteristic, and hence, every subgroup is automorph-conjugate.
- Group in which every subgroup is automorph-conjugate: In a complete group, or more generally in a group in which every automorphism is inner,
*every*subgroup is automorph-conjugate. Examples include the symmetric groups of degree , .`Further information: Symmetric groups are complete`

### Low-occurrence examples

- In an abelian group, and more generally, in a Dedekind group, every subgroup is normal, and hence, every automorph-conjugate subgroup is characteristic.
- Group in which every automorph-conjugate subgroup is characteristic: Many groups occurring in practice have this property. For instance, any group occurring as a Frattini-embedded normal subgroup in a bigger group.
`Further information: Frattini-embedded normal-realizable implies ACIC`

### Miscellaneous examples

- Sylow subgroups in finite groups are automorph-conjugate.
`Further information: Sylow implies automorph-conjugate` - In a free group on two generators, the cyclic subgroup generated by the commutator of the two generators is automorph-conjugate.
`Further information: Subgroup generated by commutator of generators of free group on two generators is automorph-conjugate`

## Metaproperties

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Metaproperty name | Satisfied? | Proof | Difficulty level (0-5) | Statement with symbols |
---|---|---|---|---|

transitive subgroup property | Yes | automorph-conjugacy is transitive | If are groups such that is automorph-conjugate in and is automorph-conjugate in , then is automorph-conjugate in . | |

trim subgroup property | Yes | Obvious reasons | 0 | and are characteristic in |

finite-intersection-closed subgroup property | No | automorph-conjugacy is not finite-intersection-closed | We can have a group and automorph-conjugate subgroups of such that the intersection is not automorph-conjugate. | |

finite-join-closed subgroup property | No | automorph-conjugacy is not finite-join-closed | We can have a group and automorph-conjugate subgroups of such that the join is not automorph-conjugate. | |

quotient-transitive subgroup property | Yes | automorph-conjugacy is quotient-transitive | If , with automorph-conjugate and normal in , and automorph-conjugate in , then is automorph-conjugate in . | |

intermediate subgroup condition | No | automorph-conjugacy does not satisfy intermediate subgroup condition | We can have such that is automorph-conjugate in but is not automorph-conjugate in . | |

centralizer-closed subgroup property | Yes | automorph-conjugacy is centralizer-closed | If is automorph-conjugate in , so is its centralizer . | |

normalizer-closed subgroup property | Yes | automorph-conjugacy is normalizer-closed | If is automorph-conjugate in , so is its normalizer . |

## Relation with other properties

### Stronger properties

### Weaker properties

### Incomparable properties

Property | Meaning | Proof of failure of implication | Proof of failure of reverse implication |
---|---|---|---|

Hall subgroup | subgroup whose order and index are relatively prime | Hall not implies automorph-conjugate | take any characteristic non-Hall subgroup, e.g., Z2 in Z4. |

normal subgroup | equals all its conjugate subgroups | Example: Z2 in V4 | Example: S2 in S3 |

### Related group properties

## Effect of property operators

Operator | Meaning | Result of application | Proof and/or additional observations |
---|---|---|---|

intermediately operator | automorph-conjugate in every intermediate subgroup | intermediately automorph-conjugate subgroup | stronger than pronormal subgroup |

join-transiter | join with any automorph-conjugate subgroup is automorph-conjugate | join-transitively automorph-conjugate subgroup | |

intersection-transiter | intersection with any automorph-conjugate subgroup is automorph-conjugate | intersection-transitively automorph-conjugate subgroup |

## Testing

### GAP code

GAP-codable subgroup propertyOne can write code to test this subgroup property inGAP (Groups, Algorithms and Programming), though there is no direct command for it.

View other GAP-codable subgroup properties | View subgroup properties with in-built commands

Here is a short piece of code that can be used to test whether a subgroup in a finite group is automorph-conjugate. The code is not very efficient.

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;;

## References

### Journal references

- Paper:Wielandt58
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