Weakly pronormal subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed weakly pronormal or is said to satisfy the Frattini property if it satisfies the following equivalent conditions:


 * Given any $$g \in G$$, there exists $$x \in H^{\langle g \rangle}$$ such that $$H^x = H^g$$. Here $$H^{\langle g \rangle}$$ denotes the smallest subgroup of $$G$$ containing $$H$$ which is closed under conjugation by $$g$$.
 * If $$H \le K \le L \le G$$ are such that $$K$$ is a normal subgroup of $$L$$, we have $$KN_L(H) = L$$.

Relation with other properties
For a picture of related subnormal-to-normal subgroup properties, refer this implication diagram.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Join-transitively pronormal subgroup
 * Weaker than::Pronormal subgroup
 * Weaker than::Abnormal subgroup
 * Weaker than::Weakly abnormal subgroup
 * Weaker than::Intermediately isomorph-conjugate subgroup
 * Weaker than::Intermediately automorph-conjugate subgroup
 * Weaker than::Sylow subgroup
 * Weaker than::Sylow subgroup of normal subgroup: This follows from Sylow of normal implies pronormal and pronormal implies weakly pronormal.
 * Weaker than::Intermediately isomorph-conjugate subgroup of normal subgroup
 * Weaker than::Intermediately automorph-conjugate subgroup of normal subgroup
 * Weaker than::Weakly procharacteristic subgroup

Weaker properties

 * Stronger than::Polynormal subgroup
 * Stronger than::Intermediately subnormal-to-normal subgroup
 * Stronger than::Subnormal-to-normal subgroup
 * Stronger than::Subgroup with weakly abnormal normalizer
 * Stronger than::Subgroup with self-normalizing normalizer