Group in which every automorph-conjugate subgroup is characteristic
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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The version of this for finite groups is at: finite ACIC-group
Definition
A group is termed a 'group in which every automorph-conjugate subgroup is characteristic if it satisfies the following equivalent conditions:
- Every automorph-conjugate subgroup of it is characteristic
- Every automorph-conjugate subgroup is normal.
Formalisms
In terms of the subgroup property collapse operator
This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (automorph-conjugate subgroup) satisfies the second property (characteristic subgroup), and vice versa.
View other group properties obtained in this way
The property of being an ACIC-group can be viewed as the collapse:
Automorph-conjugate subgroup = Characteristic subgroup
Relation with other properties
Stronger properties
- Abelian group
- Finite-Frattini-realizable group
- Frattini-embedded normal-realizable group: For full proof, refer: Frattini-embedded normal-realizable implies ACIC
- Dedekind group
- Hereditarily ACIC-group
Weaker properties
- Nilpotent group (for finite groups): This follows from the fact that Sylow subgroups are automorph-conjugate. For full proof, refer: ACIC implies nilpotent (finite groups)The implication does not hold for infinite groups. For full proof, refer: ACIC not implies nilpotent (infinite groups). Also, the converse implication does not hold even for finite groups. For full proof, refer: Nilpotent not implies ACIC
- ACIC-embeddable group
Metaproperties
Metaproperty | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | No | ACIC is not subgroup-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() |
quotient-closed group property | No | ACIC is not quotient-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() |
normal subgroup-closed group property | No | ACIC is not normal subgroup-closed | It is possible to have a group ![]() ![]() ![]() ![]() ![]() |
finite direct product-closed group property | No | ACIC is not finite direct product-closed | It is possible to have groups ![]() ![]() ![]() ![]() ![]() |
characteristic subgroup-closed group property | Yes | ACIC is characteristic subgroup-closed | Suppose ![]() ![]() ![]() ![]() |
characteristic quotient-closed group property | Yes | ACIC is characteristic quotient-closed | Suppose ![]() ![]() ![]() ![]() |
Testing
GAP code
One can write code to test this group property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View other GAP-codable group properties | View group properties with in-built commands
The following GAP code can be used to check whether a group is ACIC:
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;; IsACIC := function(G) local H; if IsAbelian(G) then return true; fi; for H in List(ConjugacyClassesSubgroups(G),Representative) do if IsAutomorphConjugateSubgroup(G,H) and not IsNormal(G,H) then return false; fi; od; return true; end;;
To do the test, enter:
IsACIC(G)
where is the group that needs to be tested. The code works only for finite groups, and as such, is extremely inefficient.