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On the theory of soluble factorizable groups

Published online by Cambridge University Press:  17 April 2009

Otto-Uwe Kramer
Otto-Uwe Kramer
Affiliation:
Mathematisches Seminar der Christian-Albrechts-Universität, Kiel, Germany.
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Abstract

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Suppose that a finite soluble group G is the product AB of subgroups A and B. Our question is the following: what conclusions can be made about G if A and B are suitably restricted? First we shall prove that the p–length of G is restricted by the derived lengths of the Sylow p–subgroups of A and B, if A and B are p–closed and p′-closed. Moreover, if in such a group the Sylow p–subgroups of A and B are modular, the p–length of G is at most 1. Next we obtain a general estimate for the derived length of the group G = AB of odd order in terms of the derived lengths of A and B. Furthermore it will be possible to bound the nilpotent length of G and also the p–length of G in terms of other invariants of special subgroups of G.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 1 , August 1976 , pp. 97 - 110
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Gross, Fletcher, “Finite groups which are the product of two nilpotent subgroups”, Bull. Austral. Math. Soc. 9 (1973), 267–274.CrossRefGoogle Scholar
[2]Hall, P. and Higman, Graham, “On the p–length of p–soluble groups and reduction theorems for Burnside's problem”, Proc. London Math. Soc. (3) 6 (1956), 1–42.CrossRefGoogle Scholar
[3]Huppert, B., Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[4]Kegel, Otto H., “Produkte nilpotenter Gruppen”, Arch. Math. (Basel) 12 (1961), 90–93.CrossRefGoogle Scholar
[5]Maier, Rudolf, “Endliche metanilpotente Gruppen”, Arch. Math. (Basel) 23 (1972), 139–144.CrossRefGoogle Scholar
[6]Pennington, Elizabeth, “Trifactorisable groups”, Bull. Austral. Math. Soc. 8 (1973), 461–469.CrossRefGoogle Scholar
[7]Ward, J.N., “On finite soluble groups”, J. Austral. Math. Soc. 9 (1969), 250–251.CrossRefGoogle Scholar