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 1996 and arose in the context of the proof that for finite groups, every quasinormal subgroup is a subnormal subgroup.
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.
See also Quasinormal subgroup