Self-conjugate-permutable subgroup

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History

Origin

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

Definition

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 $H^g = gHg^{-1}$.

Relation with other properties

Stronger properties

• Normal subgroup
• Maximal subgroup
• Abnormal subgroup
• Pronormal subgroup: For proof of the implication, refer Pronormal implies self-conjugate-permutable and for proof of its strictness (i.e. the reverse implication being false) refer Self-conjugate-permutable not implies pronormal.

Opposite properties

• Conjugate-permutable subgroup: In fact, a subgroup is both conjugate-permutable and self-conjugate-permutable iff it is normal. For full proof, refer: Conjugate-permutable and self-conjugate-permutable implies normal

References

• Groups with conjugate-permutable conditions by Shirong Li and Zhongchuan Meng, Mathematical Proceedings of the Royal Irish Academy