Contranormal subgroup
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 an opposite of normality
Contents
History
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Definition
Symbol-free definition
A subgroup of a group is contranormal if it satisfies the following equivalent conditions:
- Its normal closure (i.e. the smallest normal subgroup containing it) in the group is the whole group
- There is no proper subgroup of the whole group, containing every conjugate of the given subgroup
Definition with symbols
A subgroup of a group is termed contranormal in if the normal closure is equal to .
Note that this is not equivalent to saying that every element of is conjugate to an element of : that property is termed being a conjugate-dense subgroup.
Every maximal subgroup is either normal or contranormal.
Formalisms
Monadic second-order description
This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups
View other monadic second-order subgroup properties
A subgroup in a group is contranormal if it satisfies the following monadic second-order condition:
We are essentially using the fact that the normal closure has a monadic second-order description.
Relation with other properties
Stronger properties
- Non-normal maximal subgroup
- Abnormal subgroup
- Weakly abnormal subgroup
- Strongly contranormal subgroup
- Conjugate-dense subgroup
Weaker properties
- Cocommutatorial subgroup: A subgroup which along with the commutator subgroup generates the whole group
Incomparable properties
- Self-normalizing subgroup: Though these are closely related, neither implies the other. For full proof, refer: Self-normalizing not implies contranormal, contranormal not implies self-normalizing
- Core-free subgroup: Though these are closely related, neither implies the other. This is easily observed from the fact that core-freeness is a notion of being small while self-normalizing is a notion of being big.
Facts
The descendant-contranormal factorization
Every subgroup of a group can be expressed as a contranormal subgroup of a descendant subgroup. For a subgroup , each term of the descending serise is the normal closuer of inside its predecessor.
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: View variations of this property that are transitive | View variations of this property that are not transitive
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 and each is contravariant in the next, then is contranormal in . The proof of this follows from the fact that the normal closure of in can be obtained by first taking the normal closure of in , and then again of in .
Upward-closedness
This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties
Any subgroup containing a contranormal subgroup is contranormal. This follows from the fact that the normal closure of a bigger subgroup contains the normal closure of a smaller subgroup.
Intermediate subgroup condition
Contranormality does not satisfy the intermediate subgroup condition. It seems possible that every subgroup is potentially contranormal, though a proof is not immediate.
NCI
This subgroup property is a NCI-subgroup property, i.e., it is identity-true subgroup property and further, the only normal subgroup of a group that satisfies the property is the whole group
The only normal contranormal subgroup of a group is the whole group.
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 contranormal subgroups need not be contranormal. This follows from the fact that contranormality is an NCI-subgroup property.
Effect of property operators
The intermediately operator
Applying the intermediately operator to this property gives: weakly abnormal subgroup
If is a subgroup such that is contranormal in every intermediate subgroup , then is termed a weakly abnormal subgroup of .
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 the GAP code for testing this subgroup property at: IsContranormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
While there is no built-in GAP command for testing contranormality, this can be accomplished by a short piece of GAP code, available at GAP:IsContranormal. The command is invoked as follows:
IsContranormal(group,subgroup);
References
- Nilpotent subgroups of finite soluble groups by John S. Rose, Math. Zeitschr. 106, 97-112 (1968)
- Abnormal, pronormal, contranormal and Carter subgroups in some generalized minimax groups by L.A. Kurdachenko, J. Otal and I.Ya. Subbotin, Commun. Algebra 33, No.12, 4595-4616 (2005)
External links
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