Automorph-conjugate subgroup

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


 * 1) Any Defining ingredient::automorph (i.e. any subgroup to which it can go via an automorphism of the whole group), is also conjugate to the subgroup.
 * 2) Whenever the bigger group is embedded as a Defining ingredient::normal subgroup of some ambient group, the product of the bigger group with the Defining ingredient::normalizer of the smaller group in the ambient group, is the whole group.
 * 3) 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:


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

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

Formalisms
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.

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 $$n$$, $$n \ne 6$$.

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.

Miscellaneous examples

 * Sylow subgroups in finite groups are automorph-conjugate.
 * In a free group on two generators, the cyclic subgroup generated by the commutator of the two generators is automorph-conjugate.

Stronger properties

 * Weaker than::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.
 * Weaker than::Isomorph-conjugate subgroup
 * Weaker than::Characteristic subgroup:
 * Weaker than::Sylow subgroup:
 * Weaker than::Intermediately automorph-conjugate subgroup
 * Weaker than::Join-transitively automorph-conjugate subgroup
 * Weaker than::Intersection-transitively automorph-conjugate subgroup

Weaker properties

 * Stronger than::Intersection of automorph-conjugate subgroups
 * Stronger than::Join of automorph-conjugate subgroups
 * Stronger than::Core-characteristic subgroup
 * Stronger than::Closure-characteristic subgroup
 * Stronger than::Normal-to-characteristic subgroup

Incomparable properties

 * Hall subgroup:

Related group properties

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

Metaproperties
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 $$G$$.

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.

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

A join of automorph-conjugate subgroups need not be automorph-conjugate.

The centralizer of an automorph-conjugate subgroup of a group is again automorph-conjugate.

The normalizer of an automorph-conjugate subgroup of a group is again automorph-conjugate.

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
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;;