Paranormal subgroup

Definition with symbols
(right-action convention): A subgroup $$H$$ of a group $$G$$ is termed paranormal if for any $$g \in G$$, $$H$$ is a defining ingredient::contranormal subgroup of $$\langle H, H^g \rangle$$; in other words, the normal closure of $$H$$ in $$\langle H, H^g \rangle$$ is $$\langle H, H^g \rangle$$.

Here $$H^g = g^{-1}Hg$$ is a conjugate of $$H$$, and the angled braces are for the subgroup generated.

(left-action convention): A subgroup $$H$$ of a group $$G$$ is termed paranormal if for any $$g \in G$$, $$H$$ is a defining ingredient::contranormal subgroup of $$\langle H, gHg^{-1} \rangle$$; in other words, the normal closure of $$H$$ in $$\langle H, gHg^{-1} \rangle$$ is $$\langle H, gHg^{-1} \rangle$$.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Pronormal subgroup:
 * Weaker than::Abnormal subgroup
 * Weaker than::Join of pronormal subgroups
 * Weaker than::Strongly paranormal subgroup:
 * Weaker than::Weakly abnormal subgroup:
 * Weaker than::Paracharacteristic subgroup
 * Weaker than::Sylow subgroup
 * Weaker than::Sylow subgroup of normal subgroup
 * Weaker than::Hall subgroup:
 * Weaker than::Hall subgroup of normal subgroup:
 * Weaker than::Intermediately isomorph-conjugate subgroup
 * Weaker than::Procharacteristic subgroup

Weaker properties

 * Stronger than::Polynormal subgroup: It has been conjectured that for finite groups, the two notions coincide; however this has neither been proved nor disproved.
 * Stronger than::Weakly normal subgroup:
 * Stronger than::Intermediately subnormal-to-normal subgroup
 * Stronger than::Subnormal-to-normal subgroup

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

Testing
There is no built-in GAP command to test paranormality, but a short piece of GAP code can achieve this. The code is available at GAP:IsParanormal.