‎4‎‎-quasinormal subgroups of prime order

Generalizing the concept of quasinormality, a subgroup h of a group g is said to be 4- quasinormal in g if, for all cyclic subgroups k of g, ⟨h, k⟩ = hkhk. an intermediate concept would be 3-quasinormality, but in finite p-groups - our main concern - this is equivalent to quasinormality. quasinormal subgroups have many interesting properties and it has been shown that some of them can be extended to 4-quasinormal subgroups, particularly in finite p-groups. however, even in the smallest case, when h is a 4-quasinormal subgroup of order p in a finite p-group g, precisely how h is embedded in g is not immediately obvious. here we consider one of these questions regarding the commutator subgroup [h, g].