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Automorph-conjugate subgroup
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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.
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This is a variation of characteristicity
Find other variations of characteristicity | 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
such that σ(H) = gHg − 1).
- Whenever
for some group M, GNM(H) = M.
- Suppose A is a generating set for the automorphism group
. Then, σ(H) is a conjugate subgroup of H for every
.
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 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 automorph
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,
. 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
- Intersection of automorph-conjugate subgroups
- Join of automorph-conjugate subgroups
- Core-characteristic subgroup
- Closure-characteristic subgroup
- Normal-to-characteristic subgroup
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 in the whole group.
ABOUT THIS PROPERTY: |
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 other trim subgroup properties | View other trivially true subgroup properties | View other 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
NO: This subgroup property does not satisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some intermediate subgroup.
ABOUT THIS PROPERTY: |
ABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about 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 a join 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 other 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;;