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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

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