‎‎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].