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 of a group is termed self-conjugate-permutable if whenever is such that , then . Here .
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