Self-conjugate-permutable subgroup

Origin
The notion of self-conjugate-permutable subgroup was introduced by Shirong Li and Zhongchuan Meng, in their paper Groups with Conjugate-permutable conditions.

Symbol-free definition
A subgroup of a group is termed self-conjugate-permutable if the only conjugate subgroup with which it permutes, is itself.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed self-conjugate-permutable if whenever $$g \in G$$ is such that $$HH^g = H^gH$$, then $$H^g = H$$. Here $$H^g = gHg^{-1}$$.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Maximal subgroup
 * Weaker than::Abnormal subgroup
 * Weaker than::Pronormal subgroup:

Opposite properties

 * Conjugate-permutable subgroup: In fact, a subgroup is both conjugate-permutable and self-conjugate-permutable iff it is normal.