Contranormal subgroup

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
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.

Stronger properties

 * Non-normal maximal subgroup
 * Weaker than::Abnormal subgroup
 * Weaker than::Weakly abnormal subgroup
 * Weaker than::Strongly contranormal subgroup
 * Weaker than::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.
 * 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 $$H$$, each term of the descending serise is the normal closuer of $$H$$ inside its predecessor.

Metaproperties
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 $$K$$.

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.

The only normal contranormal subgroup of a group is the whole group.

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
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
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);

Article links

 * Preprint of the paper on abnormal, pronormal, contranormal and Carter subgroups