A note on Hamiltonian \(2\)-groups
A nonabelian finite group is called a Hamiltonian group if every subgroup is normal. A Hamiltonian 2-group is a direct product of a quaternion group of order 8 and an elementary abelian 2-group. Denote by \(H_ n\) a Hamiltonian group of order \(2^{n + 2}\). The authors study how a Hamiltonian 2-group occurs as a normal section of a 2-group. Let \(\beta(G)\) be the subgroup of \(G\) generated by \([S,G]\), where \(S\) runs over all non-normal subgroups of \(G\). It is shown that if \(G/\beta(G)\) is Hamiltonian, then \(\beta(G) = 1\). Another result is that for \(c \geq 2^ m\) there is a 2-group whose \(m\)-th derived subgroup is isomorphic to \(H_ c\). The authors also obtain the upper bound of the derived length of a 2-group which contains a normal Hamiltonian subgroup.
scientific article; zbMATH DE number 60487