Automorph-conjugate subgroup

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 $H$ of a group $G$ is termed automorph-conjugate or intravariant if... (right-action convention)	A subgroup $H$ of a group $G$ 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 $\sigma \in \operatorname{Aut}(G)$, there exists $g \in G$ such that $H^\sigma = H^g$.	for every $\sigma \in \operatorname{Aut}(G)$, there exists $g \in G$ such that $\sigma(H) = gHg^{-1}$.
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 $M$ containing $G$ as a normal subgroup, we have $GN_M(H) = M$.	(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 $A$ of $\operatorname{Aut}(G)$), we have that for any $\sigma \in A$, there exists $g \in G$ such that $H^\sigma = H^g$.	(choose a generating set $A$ of $\operatorname{Aut}(G)$), we have that every $\sigma \in A$, there exists $g \in G$ such that $\sigma(H) = gHg^{-1}$.

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

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History

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

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 $\implies$ 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.
• 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 $H\le K \le G$ are groups such that $H$ is automorph-conjugate in $K$ and $K$ is automorph-conjugate in $G$, then $H$ is automorph-conjugate in $G$.
trim subgroup property	Yes	Obvious reasons	0	$\{ e \}$ and $G$ are characteristic in $G$
finite-intersection-closed subgroup property	No	automorph-conjugacy is not finite-intersection-closed		We can have a group $G$ and automorph-conjugate subgroups $H, K$ of $G$ such that the intersection $H \cap K$ is not automorph-conjugate.
finite-join-closed subgroup property	No	automorph-conjugacy is not finite-join-closed		We can have a group $G$ and automorph-conjugate subgroups $H, K$ of $G$ such that the join $\langle H, K \rangle$ is not automorph-conjugate.
quotient-transitive subgroup property	Yes	automorph-conjugacy is quotient-transitive		If $H \le K \le G$, with $H$ automorph-conjugate and normal in $G$, and $K/H$ automorph-conjugate in $G/H$, then $K$ is automorph-conjugate in $G$.
intermediate subgroup condition	No	automorph-conjugacy does not satisfy intermediate subgroup condition		We can have $H \le K \le G$ such that $H$ is automorph-conjugate in $G$ but $H$ is not automorph-conjugate in $K$.
centralizer-closed subgroup property	Yes	automorph-conjugacy is centralizer-closed		If $H$ is automorph-conjugate in $G$, so is its centralizer $C_G(H)$.
normalizer-closed subgroup property	Yes	automorph-conjugacy is normalizer-closed		If $H$ is automorph-conjugate in $G$, so is its normalizer $N_G(H)$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
order-dominating subgroup	for any subgroup whose order divides the subgroup's order, it contains a conjugate of that subgroup			|FULL LIST, MORE INFO
order-conjugate subgroup	conjugate to any other subgroup of the group having the same order			|FULL LIST, MORE INFO
isomorph-conjugate subgroup	conjugate to any other subgroup that is isomorphic to it.			|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms	characteristic implies automorph-conjugate	automorph-conjugate not implies characteristic	Intersection-transitively automorph-conjugate subgroup, Join-transitively automorph-conjugate subgroup, Procharacteristic subgroup, Weakly procharacteristic subgroup|FULL LIST, MORE INFO
Sylow subgroup	subgroup of maximal prime power order in finite group	Sylow implies automorph-conjugate		Characteristic subgroup of Sylow subgroup, Intermediately automorph-conjugate subgroup, Order-conjugate subgroup|FULL LIST, MORE INFO
intermediately automorph-conjugate subgroup	automorph-conjugate in every intermediate subgroup		automorph-conjugacy does not satisfy intermediate subgroup condition	Weakly procharacteristic subgroup|FULL LIST, MORE INFO
join-transitively automorph-conjugate subgroup	its join (the subgroup generated by the union) with any automorph-conjugate subgroup is automorph-conjugate.		automorph-conjugacy is not finite-join-closed	|FULL LIST, MORE INFO
intersection-transitively automorph-conjugate subgroup	its intersection with any automorph-conjugate subgroup is automorph-conjugate.		automorph-conjugacy is not finite-intersection-closed	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
automorph-dominating subgroup	every automorph is contained in a conjugate			|FULL LIST, MORE INFO
intersection of automorph-conjugate subgroups	can be expressed as an intersection of automorph-conjugate subgroups of the whole group	(obvious)	automorph-conjugacy is not finite-intersection-closed	|FULL LIST, MORE INFO
join of automorph-conjugate subgroups	can be expressed as a join of automorph-conjugate subgroups of the whole group	(obvious)	automorph-conjugacy is not finite-join-closed	|FULL LIST, MORE INFO
core-characteristic subgroup	the normal core of the subgroup in the whole group is characteristic in the whole group	(via automorph-dominating)	(via automorph-dominating)	Automorph-dominating subgroup, Intersection of automorph-conjugate subgroups|FULL LIST, MORE INFO
closure-characteristic subgroup	the normal closure of the subgroup in the whole group is characteristic in the whole group	(via automorph-dominating)	(via automorph-dominating)	Automorph-dominating subgroup, Join of automorph-conjugate subgroups|FULL LIST, MORE INFO
normal-to-characteristic subgroup	if the subgroup is normal, it is also characteristic	(obvious)	any non-normal non-automorph-conjugate subgroup suffices	Join of automorph-conjugate subgroups|FULL LIST, MORE INFO

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

• Group in which every automorph-conjugate subgroup is characteristic
• Group in which every subgroup is automorph-conjugate

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

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

GAP-codable subgroup property

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}