Self-normalizing subgroup

Symbol-free definition
A subgroup of a group is termed self-normalizing if it equals its own normalizer in the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed self-normalizing if $$N_G(H) = H$$.

Formalisms
This is essentially because the normalizer of a subgroup has a first-order description.

Stronger properties

 * Weaker than::Abnormal subgroup
 * Weaker than::Weakly abnormal subgroup
 * Weaker than::Free factor: Any nontrivial free factor of a group is either self-normalizing or trivial.

Weaker properties

 * Stronger than::Central factor of normalizer
 * Stronger than::Subgroup with canonical Abelianization:
 * Stronger than::Self-centralizing subgroup:

Incomparable properties

 * Contranormal subgroup:

Metaproperties
A self-normalizing subgroup of a self-normalizing subgroup need not be self-normalizing.

Let $$G \le H \le K$$ be groups. Then the condition that $$G$$ is self-normalizing in $$K$$ means that $$N_K(G) = G$$ which will imply that $$N_H(G) = G$$, and hence that $$G$$ is self-normalizing in $$H$$.

Thus, any self-normalizing subgroup is also self-normalizing in every intermediate subgroup.

It is clear that a subgroup that is both normal and self-normalizing must be the whole group -- that's because its normalizer equals both itself and the whole group.

An intersection of self-normalizing subgroups need not be self-normalizing. This follows from the fact that it is a NCI-subgroup property, and hence cannot be normal core-closed.

A join of self-normalizing subgroups need not be self-normalizing. This follows because the property of being self-normalizing is not normal closure-closed: there exist self-normalizing subgroups whose normal closure is a proper normal subgroup.

If $$H$$ is a self-normalizing subgroup of $$G$$, and $$f:G \to K$$ is a surjective homomorphism of groups, then $$f(H)$$ is a self-normalizing subgroup of $$K$$.

If $$H_1$$ is a self-normalizing subgroup of math>G_1, and $$H_2$$ is a self-normalizing subgroup of $$G_2$$, then $$H_1 \times H_2$$ is a self-normalizing subgroup of $$G_1 \times G_2$$. The analogous statement holds for arbitrary direct products as well.

Effect of property operators
If $$H$$ is a subgroup of $$G$$ such that every subgroup of $$G$$ containing $$H$$ is self-normalizing in $$G$$, then $$H$$ is termed a weakly abnormal subgroup of $$G$$. Being weakly abnormal is also equivalent to being contranormal in every intermediate subgroup.