Abstract
Finiteness conditions associated with subnormal subgroups are in general fairly difficult to handle. In this note we refer in particular to two restrictions of this type. The first is the so-called subnormal intersection property, which demands that the intersection of any family of subnormal subgroups should again he a subnormal subgroup. The second condition entails the existence of an upper bound for the subnormal indices of all subnormal subgroups. Since their introduction by Robinson almost ten years ago, these conditions have prompted a number of investigations aimed at elucidating the structure of groups (usually soluble) with such nations on the behaviour of their subnormal subgroups. In his recent treatise [7, 8] — to which we refer the reader for background and terminology — Robinson remarks on the apparent difficulty of such investigations. Cases in point are Robinson’s proof, in [5], that finitely generated soluble groups with the subnormal intersection property are finite-by-nilpotent, and McDougall’s relatively incomplete description of soluble minimax groups with the subnormal intersection property ([4]).
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D.J. McCaughan, “Subnormal structure in some classes cf infinite groups”, Austral. Math. Soc. B (1973), 137–150. Zb1.248.20035.
D.J. McCaughan, “Subnormality in soluble minimax groups”, J. Austral. Math. Soc. (to appear).
D.J. McCaughan and D. McDougall, “The subnormal structure of metanilpotent groups”, Bull. Austral. Math. Soc. 6 (1972), 287–306. MR45#2023.
David McDougall, “Soluble minimax groups with the subnormal intersection property”, Math. Z. 114 (1970), 241–244. MR4115490.
Derek S. Robinson, “On finitely generated soluble groups”, Proc. London Math. Soc. (3) 15 (1965), 508–516. MR31#253.
Derek S. Robinson, “On the theory of subnormal subgroups”, Math. Z. 89 (1965), 30–51. MR32l2481.
Derek J.S. Robinson, Finiteness conditions and generalized soluble groups, Part 1 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 62. Springer-Verlag, Berlin, Heidelberg, New York, 1972 ). Z61.243.20032.
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 ). Zó1.243.20033.
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© 1974 Springer-Verlag Berlin Heidelberg
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McCaughan, D.J. (1974). On Subnormality in Soluble Minimax 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_45
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DOI: https://doi.org/10.1007/978-3-662-21571-5_45
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