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^[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

1. Foguel, Tuval (1997), "Conjugate-permutable subgroups", Journal of Algebra 191 (1): 235–239, doi:10.1006/jabr.1996.6924, MR 1444498.