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Zbl 0799.20023
Bryce, R.A.; Cossey, John
A note on Hamiltonian $2$-groups. (English)
[J] Rend. Semin. Mat. Univ. Padova 86, 175-182 (1991). ISSN 0041-8994

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 $H\sb n$ a Hamiltonian group of order $2\sp{n + 2}$. 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 $c \geq 2\sp m$ there is a 2-group whose $m$-th derived subgroup is isomorphic to $H\sb c$. The authors also obtain the upper bound of the derived length of a 2-group which contains a normal Hamiltonian subgroup.
[H.Yamada (Tokyo)]

MSC 2000:: *20D15 Nilpotent finite groups
20D30 Series and lattices of subgroups of finite groups
20D25 Special subgroups of finite groups
20E07 Subgroup theorems (group theory)

Keywords: finite group; Hamiltonian 2-group; quaternion group; elementary abelian 2-group; normal section; derived subgroup; derived length

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