On the structure of generalized Hamiltonian groups
A group \(G\) is said to be a generalized Dedekind group if the subgroup generated by all non-normal subgroups of \(G\) is properly contained in \(G\). Generalized Dedekind groups were introduced and studied by \textit{D. Cappitt} [J. Algebra 17, 310-316 (1971; Zbl 0232.20067)]. If \(G\) is a group, let \(CS(G)\) denote the subgroup generated by all cyclic non-normal subgroups of \(G\); it was proved by \textit{D. M. Reboli} [Groups St. Andrews 1997, Lond. Math. Soc. Lect. Note Ser. 261, 624-632 (1999; Zbl 0936.20028)] that for any group \(G\) the subgroup \(CS(G)\) coincides with the subgroup generated by all non-normal subgroups of \(G\). A group \(G\) is called a generalized Hamiltonian group if \(\{1\}\neq CS(G)\neq G\). It is proved in this article that if \(G\) is a generalized Hamiltonian group, then the commutator subgroup \(G'\) of \(G\) is cyclic and either \(|G'|=p^k\) or \(|G'|=2p^k\), where \(p\) is a prime number and \(k\) is a positive integer.
scientific article; zbMATH DE number 1556583