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
- 1 History
- 2 Definition
- 3 Formalisms
- 4 Relation with other properties
- 5 Facts
- 6 Metaproperties
- 7 Effect of property operators
- 8 Testing
- 9 References
- 10 External links
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- 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
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.
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
- Non-normal maximal subgroup
- Abnormal subgroup
- Weakly abnormal subgroup
- Strongly contranormal subgroup
- Conjugate-dense subgroup
- Cocommutatorial subgroup: A subgroup which along with the commutator subgroup generates the whole group
- 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.
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.
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 .
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.
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.
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
If is a subgroup such that is contranormal in every intermediate subgroup , then is termed a weakly abnormal subgroup of .
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:
- 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)
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