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 ![]() ![]() |
A subgroup ![]() ![]() |
---|---|---|---|---|
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 ![]() ![]() ![]() |
for every ![]() ![]() ![]() |
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 ![]() ![]() ![]() |
(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 ![]() ![]() ![]() ![]() ![]() |
(choose a generating set ![]() ![]() ![]() ![]() ![]() |
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
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 conjugate subgroups. In other words,
is automorph-conjugate in
iff for every automorph
of
,
and
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
,
. 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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
trim subgroup property | Yes | Obvious reasons | 0 | ![]() ![]() ![]() |
finite-intersection-closed subgroup property | No | automorph-conjugacy is not finite-intersection-closed | We can have a group ![]() ![]() ![]() ![]() | |
finite-join-closed subgroup property | No | automorph-conjugacy is not finite-join-closed | We can have a group ![]() ![]() ![]() ![]() | |
quotient-transitive subgroup property | Yes | automorph-conjugacy is quotient-transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
intermediate subgroup condition | No | automorph-conjugacy does not satisfy intermediate subgroup condition | We can have ![]() ![]() ![]() ![]() ![]() | |
centralizer-closed subgroup property | Yes | automorph-conjugacy is centralizer-closed | If ![]() ![]() ![]() | |
normalizer-closed subgroup property | Yes | automorph-conjugacy is normalizer-closed | If ![]() ![]() ![]() |
Relation with other properties
Stronger properties
Weaker properties
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.GAP-codable subgroup property
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;;