Groups with few non-quasinormal subgroups (Q2366596)

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 \textit{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 \(G\). The author studies the class \(\mathbf X\) 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 \textit{D. Cappitt} [J. Algebra 17, 310-316 (1971; Zbl 0232.20067)]. Clearly every \(\mathbf X\)-group is generated by cyclic quasinormal subgroups, and in particular it is locally nilpotent. The author proves that non-periodic \(\mathbf X\)-groups are quasi-Hamiltonian. The investigation of periodic \(\mathbf X\)-groups can be reduced to the case of a \(p\)-group (\(p\) prime), and the description of \(p\)-groups in the class \(\mathbf X\) is obtained. In particular, it is shown that a \(p\)-group of infinite exponent \(G\) is in the class \(\mathbf X\) 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 \(\mathbf X\)-group whose Sylow 2-subgroup is quasi-Hamiltonian, then \(G\) is metabelian.