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A note on Hamiltonian \(2\)-groups. (English) Zbl 0799.20023

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.
Reviewer: H.Yamada (Tokyo)

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20D30 Series and lattices of subgroups
20D25 Special subgroups (Frattini, Fitting, etc.)
20E07 Subgroup theorems; subgroup growth
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References:

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