# 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;;