Abnormal subgroup

Origin
The notion of abnormal subgroup was introduced by Roger W. Carter in his attempts to understand the structure of Carter subgroups of a solvable group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed abnormal if it satisfies the following equivalent conditions:
 * 1) * (Right action convention): For any $$x$$ in $$G$$, $$x$$ lies inside the subgroup $$\langle H,H^x \rangle$$. Here $$H^x$$ denotes the conjugate subgroup $$x^{-1}Hx$$.
 * 2) * (Left action convention): For any $$x \in G$$, we have $$x \in \langle H, xHx^{-1} \rangle$$.
 * 3) $$H$$ is a defining ingredient::weakly abnormal subgroup of $$G$$ and is not contained in the defining ingredient::intersection of two distinct conjugate subgroups.
 * 1) $$H$$ is a defining ingredient::weakly abnormal subgroup of $$G$$ and is not contained in the defining ingredient::intersection of two distinct conjugate subgroups.

Weaker properties

 * Stronger than::Self-normalizing subgroup
 * Stronger than::Weakly abnormal subgroup
 * Stronger than::Subabnormal subgroup
 * Stronger than::Pronormal subgroup
 * Stronger than::Weakly pronormal subgroup
 * Stronger than::Paranormal subgroup
 * Stronger than::Polynormal subgroup
 * Stronger than::Contranormal subgroup

Opposites
The only subgroup of a group that is both normal and abnormal is the whole group itself.

Metaproperties
If $$H$$ is abnormal inside $$G$$, $$H$$ is also abnormal inside $$K$$ for any intermediate subgroup $$K$$.

If $$H$$ is abnormal inside $$G$$, then so is any subgroup $$K$$ of $$G$$ containing $$H$$.

The property of being abnormal is not transitive. Its subordination is the property of being subabnormal.

Trimness
The property of being abnormal is identity-true, that is, any group is abnormal as a subgroup of itself. It is not true for the trivial subgroup unless the whole group is trivial.

Testing
There's no in-built function to test for abnormality, but a short snippet of code can be used to test if a subgroup is abnormal. The function is invoked as follows:

IsAbnormal(group,subgroup);