Let $H$, $L$ and $X$ be subgroups of a finite group$G$. Then $H$ is said to be $X$-permutable with $L$ if for some$xin X$ we have $AL^{x}=L^{x}A$. We say that $H$ is emph{$X$-quasipermutable } (emph{$X_{S}$-quasipermutable}, respectively) in $G$ provided $G$ has a subgroup$B$ such that $G=N_{G}(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (with all Sylowsubgroups, respectively) $...

Abstract A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of G permutably embedded in $\langle Q,g\rangle $ for each element g . It proved here contains normal such $G/Q$ ?ernikov, then has finite index; moreover, periodic, it ?ernikov N $G/N$ quasihamiltonian. This result should be compared with theorems Schlette stating over centre, abelian-...