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Zbl 0782.20021
Mainardis, Mario
Groups with few non-quasinormal subgroups. (Italian)
[J] Rend. Semin. Mat. Univ. Padova 88, 245-261 (1992). ISSN 0041-8994

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)]

MSC 2000:: *20E15 Chains and lattices of subgroups of groups
20E34 General structure theorems of groups
20F19 Generalizations of solvable and nilpotent groups
20D40 Products of subgroups of finite groups
20E07 Subgroup theorems (group theory)
20E25 Local properties of groups
20F50 Periodic groups; locally finite groups

Keywords: quasi-hamiltonian groups; generated by cyclic quasinormal subgroups; locally nilpotent; non-periodic $\bold X$-groups; periodic $\bold X$-groups; $p$-groups; metabelian

Citations: Zbl 0061.025; Zbl 0232.20067

Cited in: Zbl 1239.20024

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