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):

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 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.

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

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
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;;