Abstract
A group is said to be hyperabelian if each of its non-trivial quotient groups has a non-trivial abelian normal subgroup, and subsoluble if each of its non-trivial quotient groups has a non-trivial abelian subnormal subgroup. In this note we settle a point raised by Robinson ([2], p. 87) by showing that subsoluble groups satisfying Min-n, the minimal condition for normal subgroups, need not be hyperabelian. More exactly, we construct a group G whose normal subgroups are well-ordered by inclusion, of order-type ω + 1, having a perfect minimal normal subgroup N which is generated by its abelian normal subgroups, such that G/N is locally soluble and hyperabelian; G is obviously a group satisfying Min-n which is subsoluble but not hyperabelian. Our construction uses the notion of the treble product rower of a family of groups introduced in [1].
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References
H. Heineken and J.S. Wilson, “Locally soluble groups with Min-n ”, J. Austral. Math. Soc (to appear).
Derek J.S. Robinson, Finiteness conditions and generalized soluble groups, Part 2 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 63. Springer-Verlag, Berlin, Heidelberg, New York, 1972). Zb1. 243. 20033.
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© 1974 Springer-Verlag Berlin Heidelberg
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Wilson, J.S. (1974). A Note on Subsoluble Groups. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_75
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DOI: https://doi.org/10.1007/978-3-662-21571-5_75
Publisher Name: Springer, Berlin, Heidelberg
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