‎$‎۴‎$‎-quasinormal subgroups of prime order

‎Generalizing the concept of quasinormality‎, ‎a subgroup $H$ of a group $G$ is said to be ۴-quasinormal in $G$ if‎, ‎for all cyclic subgroups $K$ of $G$‎, ‎$\langle H,K\rangle=HKHK$‎. ‎An intermediate concept would be ۳-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 ۴-quasinormal subgroups‎, ‎particularly in finite‎ ‎$p$-groups‎. ‎However‎, ‎even in the smallest case‎, ‎when $H$ is a ۴-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 $[H,G]$‎.

Keywords:

‎Finite group‎ , ‎Sylow subgroup‎ , ‎abnormal subgroup‎ , ‎seminormal subgroup