A subgroup $H$ of a group $G$ is called a pronormal subgroup if for all $x\in G$ the subgroups $H$ and $xHx^{-1}$ are conjugate in $\genby{H,xHx^{-1}}$ .

Some facts about pronormal subgroups:

• Normal subgroups are pronormal.
• Maximal subgroups are pronormal.
• Abnormal subgroups are pronormal.
• Sylow subgroups of finite groups are pronormal.
• The normalizer of a pronormal subgroup is abnormal.
• A pronormal subgroup is normal if and only if it is subnormal.