Malnormal subgroup

Symbol-free definition
A subgroup of a group is termed malnormal if it satisfies the following equivalent conditions:
 * 1) Its conjugate by any element outside the subgroup intersects it trivially.
 * 2) It is a self-normalizing subgroup and is also a TI-subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed malnormal if for any $$g$$ outside $$H$$, the subgroup $$gHg^{-1}$$ intersects $$H$$ trivially.

Stronger properties

 * Weaker than::Frobenius subgroup (also called Frobenius complement) is defined as a proper nontrivial malnormal subgroup in a finite group

Weaker properties

 * Stronger than::TI-subgroup
 * Stronger than::Self-normalizing subgroup

Oppositeness to normality
The only normal malnormal subgroup of a group is the whole group itself.

Metaproperties
A malnormal subgroup of a malnormal subgroup is malnormal. That is, if $$G \le H \le K$$ such that $$H$$ is malnormal in $$K$$ and $$G$$ is malnormal in $$H$$, then $$G$$ is malnormal in $$K$$. The proof relies on the fact that every element in $$K \setminus G$$ lies either in $$K \setminus H$$ or $$H \setminus G$$, and the use of malnormality in each case.

Every group is malnormal as a subgroup of itself: the condition is vacuously true because there is no element outside.

The trivial group is also always malnormal, because its intersection with anything is trivial.

if $$H$$ is a malnormal subgroup of $$G$$ and $$K$$ is any subgroup of $$G$$, then $$H \cap K$$ is clearly malnormal in $$K$$.

An arbitrary intersection of malnormal subgroups is malnormal. This follows from the fact that any element outside the intersection must lie outside at least one of the subgroups being intersected.