Gruppi con pochi sottogruppi non quasinormali

Let $G$ be a group. A subgroup $H$ of $G$ is said to be quasinormal if $HK = KH$ for every subgroup $K$ of $G$, and $G$ is called quasi-Hamiltonian if all its subgroups are quasinormal. The structure of quasi-Hamiltonian groups has been described by {\it K. Iwasawa} [in J. Fac. Sci., Univ. Tokyo, Sect. I 4, 171-199 (1941; Zbl 0061.025) and Jap. J. Math. 18, 709-728 (1943; Zbl 0061.025)]. If $G$ is a group, let $Q(G)$ denote the subgroup generated by all subgroups of $G$ which are not quasinormal. Then $G$ is quasi-Hamiltonian if and only if $Q(G) = 1$, and it is easy to show that $Q(G)$ is generated by all cyclic non-quasinormal subgroups of $G$.\par The author studies the class $\bold X$ of all groups $G$ for which $Q(G)$ is a proper subgroup. The corresponding problem for the subgroup generated by all non-normal subgroups was considered by {\it D. Cappitt} [J. Algebra 17, 310-316 (1971; Zbl 0232.20067)]. Clearly every $\bold X$-group is generated by cyclic quasinormal subgroups, and in particular it is locally nilpotent. The author proves that non-periodic $\bold X$-groups are quasi-Hamiltonian. The investigation of periodic $\bold X$-groups can be reduced to the case of a $p$-group ($p$ prime), and the description of $p$-groups in the class $\bold X$ is obtained. In particular, it is shown that a $p$-group of infinite exponent $G$ is in the class $\bold X$ if and only if the subgroup generated by all non-normal subgroups of $G$ is properly contained in $G$. Finally, the author proves that if $G$ is an $\bold X$-group whose Sylow 2-subgroup is quasi-Hamiltonian, then $G$ is metabelian. [F.de Giovanni (Napoli)]