Conjugate-permutable subgroup
In
conjugate subgroups. The term was introduced by Tuval Foguel in 1997[1] and arose in the context of the proof that for finite groups, every quasinormal subgroup is a subnormal subgroup
.
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
- MR 1444498.