Contranormal subgroup

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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

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 H of a group G is termed contranormal in G if the normal closure H^G is equal to G.

Note that this is not equivalent to saying that every element of G is conjugate to an element of H: 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 H in a group G is contranormal if it satisfies the following monadic second-order condition:

\forall A \subset G, [(x \in H, g \in G \implies gxg^{-1} \in A) \land (x,y \in A \implies xy^{-1} \in A)] \implies A = G

We are essentially using the fact that the normal closure has a monadic second-order description.

Relation with other properties

Stronger properties

Weaker properties

Incomparable properties

Facts

The descendant-contranormal factorization

Every subgroup of a group can be expressed as a contranormal subgroup of a descendant subgroup. For a subgroup H, each term of the descending serise is the normal closuer of H 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 G \le H \le K and each is contravariant in the next, then G is contranormal in K. The proof of this follows from the fact that the normal closure of G in K can be obtained by first taking the normal closure of G in H, and then again of H in K.

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 H \le G is a subgroup such that H is contranormal in every intermediate subgroup K, then H is termed a weakly abnormal subgroup of G.

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 the GAP code for testing this subgroup property at: IsContranormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
GAP-codable subgroup property

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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