Polynormal subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed polynormal if given any $$g \in G$$, $$H$$ is a contranormal subgroup in the subgroup $$H^{\langle g \rangle}$$, i.e., the closure of $$H$$ under the action by conjugation of the cyclic subgroup generated by $$g$$.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Maximal subgroup
 * Weaker than::Abnormal subgroup
 * Weaker than::Pronormal subgroup
 * Weaker than::Weakly abnormal subgroup
 * Weaker than::Weakly pronormal subgroup
 * Weaker than::Strongly paranormal subgroup
 * Weaker than::Paranormal subgroup
 * Weaker than::Strongly polynormal subgroup
 * Weaker than::Sylow subgroup in a finite group

Weaker properties

 * Stronger than::Fan subgroup
 * Stronger than::Intermediately subnormal-to-normal subgroup:

Metaproperties
If $$H$$ is polynormal in $$G$$, $$H$$ is also polynormal in any intermediate subgroup $$K$$.

The whole group and the trivial subgroup are polynormal; in fact they are normal.

In fact, an arbitrary, possibly empty, join of polynormal subgroups is polynormal.