$4$-quasinormal subgroups of prime order

Abstract

‎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 $G$‎, ‎$langle H,Krangle=HKHK$‎. ‎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 $[H,G]$‎.