Document Type: Ischia Group Theory 2018
University of Warwick
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 $G$, $\langle H,K\rangle=HKHK$. 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 $[H,G]$.