Automorph-conjugate subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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RANDOM TIP:The relation with other properties section lists stronger and weaker properties, along with links to proofs of the implications and non-implications. This helps give a feel of how the subgroup property relates to other properties.

This is a variation of characteristicity
View a complete list of variations of characteristicity OR read a survey article on varying characteristicity

History

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

This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page.

Definition

Symbol-free definition

A subgroup of a group is termed automorph-conjugate (or intravariant) if it satisfies the following equivalent conditions:

Any automorph (i.e. any subgroup to which it can go via an automorphism of the whole group), is also conjugate to the subgroup.
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.
Consider a generating set for the automorphism group of the group. Then, the image of the subgroup under any element of that generating set is conjugate to it.

Definition with symbols

A subgroup $H$ of a group $G$ is termed automorph-conjugate (or intravariant) in $G$ if it satisfies the following equivalent conditions:

For any automorphism $σ$ of $G$ , $H$ and $σ(H)$ are conjugate subgroups in $G$ (that is, there exists $g \in G$ such that $σ(H) = g H g - 1$ ).
Whenever $G \triangleleft M$ for some group $M$ , $G N M (H) = M$ .
Suppose $A$ is a generating set for the automorphism group $\operatorname{Aut}(G)$ . Then, $σ(H)$ is a conjugate subgroup of $H$ for every $\sigma \in A$ .

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

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 can be defined and viewed using a relation implication expression
View all subgroup properties having such expressions

The subgroup property of being automorph-conjugate can be expressed as automorph $\implies$ conjugate subgroups. In other words, $H$ is automorph-conjugate in $G$ iff for every automorph $K$ of $H$ , $H$ and $K$ 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 $n$ , $n \ne 6$ . 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.
ACIC-group is a group in which every automorph-conjugate subgroup is characteristic. Many groups occurring in practice are ACIC-groups. For instance, any group that occurs as a Frattini-embedded normal subgroup of a bigger group is an ACIC-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

Relation with other properties

Stronger properties

Order-dominating subgroup
- Homomorph-dominating subgroup when the subgroup is a co-Hopfian group.
- Endomorph-dominating subgroup when the subgroup is a co-Hopfian group.
Isomorph-conjugate subgroup
Characteristic subgroup: For full proof, refer: Characteristic implies automorph-conjugate
Sylow subgroup: For full proof, refer: Sylow implies automorph-conjugate
Intermediately automorph-conjugate subgroup
Join-transitively automorph-conjugate subgroup
Intersection-transitively automorph-conjugate subgroup

Weaker properties

Incomparable properties

Hall subgroup: For full proof, refer: Hall not implies automorph-conjugate

Related group properties

ACIC-group: A group in which every automorph-conjugate subgroup is characteristic.
Group in which every subgroup is automorph-conjugate

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property.
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ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties| View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

If $H$ is an automorph-conjugate subgroup of $K$ and $K$ is an automorph-conjugate subgroup of $G$ , then $H$ is an automorph-conjugate subgroup of $G$ .

For full proof, refer: Automorph-conjugacy is transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself)
View all trim subgroup properties OR view trivially true subgroup properties OR view identity-true subgroup properties

Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property
Read an article on methods to prove that a subgroup property is not intersection-closed

An intersection of automorph-conjugate subgroups need not be automorph-conjugate. In fact, an intersection of conjugate automorph-conjugate subgroups need not be automorph-conjugate either. For full proof, refer: Automorph-conjugacy is not intersection-closed, Automorph-conjugacy is not conjugate-intersection-closed

Intermediate subgroup condition

This subgroup property does not satisfy the intermediate subgroup condition

The property of being automorph-conjugate does not satisfy the intermediate subgroup condition.

Join-closedness

This subgroup property is not join-closed, viz., it is not true that an intersection of subgroups with this property must have this property
Read an article on methods to prove that a subgroup property is not join-closed

A join of automorph-conjugate subgroups need not be automorph-conjugate. For full proof, refer: Automorph-conjugacy is not join-closed

Centralizer-closedness

This subgroup property is centralizer-closed: the centralizer of any subgroup with this property, in the whole group, again has this property
View a complete list of centralizer-closed subgroup properties

The centralizer of an automorph-conjugate subgroup of a group is again automorph-conjugate. For full proof, refer: Automorph-conjugacy is centralizer-closed

Normalizer-closedness

This subgroup property is normalizer-closed: the normalizer of any subgroup with this property, in the whole group, again has this property
View a complete list of normalizer-closed subgroup properties

The normalizer of an automorph-conjugate subgroup of a group is again automorph-conjugate. For full proof, refer: Automorph-conjugacy is normalizer-closed

Effect of operators

Intermediately operator

The result of applying the intermediately operator to the property of being automorph-conjugate gives the property of being an intermediately automorph-conjugate subgroup. This implies the property of being pronormal.

Testing

GAP code

One can write code to test this subgroup property in GAP (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^{More info}

Facts about Automorph-conjugate subgroupRDF feed

Defined in	Wielandt58 (?, ?, ?) +
Defining ingredient	Automorph +, Conjugate subgroups +, Normal subgroup +, and Normalizer +
Dissatisfies metaproperty	Intermediate subgroup condition +, and Transfer condition +
Page class	Term +
Referenced in	Wielandt58 (?, ?, ?) +
Satisfies metaproperty	Transitive subgroup property +, Trim subgroup property +, Trivially true subgroup property +, and Identity-true subgroup property +
Satisfies property	Centralizer-closed subgroup property +
Stronger than	Intersection of automorph-conjugate subgroups +, Join of automorph-conjugate subgroups +, Core-characteristic subgroup +, Closure-characteristic subgroup +, and Normal-to-characteristic subgroup +
Variation of	Characteristicity +
Weaker than	Order-dominating subgroup +, Isomorph-conjugate subgroup +, Characteristic subgroup +, Sylow subgroup +, Intermediately automorph-conjugate subgroup +, Join-transitively automorph-conjugate subgroup +, and Intersection-transitively automorph-conjugate subgroup +