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

## References

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