Bryce, R. A.; Cossey, John A note on Hamiltonian \(2\)-groups. (English) Zbl 0799.20023 Rend. Semin. Mat. Univ. Padova 86, 175-182 (1991). 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) Cited in 3 Documents MSC: 20D15 Finite nilpotent groups, \(p\)-groups 20D30 Series and lattices of subgroups 20D25 Special subgroups (Frattini, Fitting, etc.) 20E07 Subgroup theorems; subgroup growth Keywords:finite group; Hamiltonian 2-group; quaternion group; elementary abelian 2-group; normal section; derived subgroup; derived length PDFBibTeX XMLCite \textit{R. A. Bryce} and \textit{J. Cossey}, Rend. Semin. Mat. Univ. Padova 86, 175--182 (1991; Zbl 0799.20023) Full Text: Numdam EuDML References: [1] R. Baer , Den Kern, eine charaykteristische Untergruppe , Compositio Math. , 1 ( 1935 ), pp. 254 - 283 . Numdam | JFM 60.0080.01 · JFM 60.0080.01 [2] R. Baer , Gruppen mit Hamiltonischen Kern , Compositio Math. , 2 ( 1935 ), pp. 241 - 246 . Numdam | JFM 61.0101.02 · JFM 61.0101.02 [3] R. Dedekind , Uber Gruppen deren sämtliche Teiler Normalteiler sind , Math. Ann. , 48 ( 1897 ), pp. 548 - 561 . Article | MR 1510943 | JFM 28.0129.03 · JFM 28.0129.03 [4] B. Huppert , Endliche Gruppen . - I. Die Grundlehren der Mathematischen Wissenschaften, Bd . 134 , Springer-Verlag , Berlin , Heidelberg , New York ( 1967 ). MR 224703 | Zbl 0217.07201 · Zbl 0217.07201 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.