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Conjugate-permutable subgroup
In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup that commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1997{{citation
Clearly, every quasinormal subgroup is conjugate-permutable.
In fact, it is true that for a finite group:
- Every maximal conjugate-permutable subgroup is normal.
- Every conjugate-permutable subgroup is a conjugate-permutable subgroup of every intermediate subgroup containing it.
- Combining the above two facts, every conjugate-permutable subgroup is subnormal.
Conversely, every 2-subnormal subgroup (that is, a subgroup that is a normal subgroup of a normal subgroup) is conjugate-permutable.
References
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