Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-04T21:09:41.365Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite dinilpotent groups of small derived length

Part of: Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

John Cossey  and
Yanming Wang
John Cossey
Affiliation:
Mathematics Department School of Mathematical Sciences Australian National UniversityCanberra 0200Australia e-mail: [email protected]
Yanming Wang
Affiliation:
Department of Mathematics Zhongshan University of Guangzhou510275 P. R.China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A finite dinilpotent group G is one that can be written as the product of two finite nilpotent groups, A and B say. A finite dinilpotent group is always soluble. If A is abelian and B is metabelian, with |A| and|B| coprime, we show that a bound on the derived length given by Kazarin can be improved. We show that G has derived length at most 3 unless G contains a section with a well defined structure: in particular if G is of odd order, G has derived length at most 3.

Keywords

finite soluble groupsdinilpotent groupsderived length

MSC classification

Secondary: 20D60: Arithmetic and combinatorial problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 3 , December 1999 , pp. 318 - 328
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Conway, J. H., Curtis, R., Norton, S., Parker, R. and Wilson, R., Atlas of finite groups (Clarendon Press, Oxford, 1985).Google Scholar
[2]Doerk, K. and Hawkes, T. O., Finite soluble groups, Expositions in Mathematics 4 (de Gruyter, Berlin, 1992).CrossRefGoogle Scholar
[3]Hall, P. and Higman, G., ‘The p-length of p-soluble groups and reduction theorems for Burnside's problem’, Proc. London Math. Soc. (3) 7 (1956), 1–42.CrossRefGoogle Scholar
[4]Higman, G., ‘Complementation of Abelian normal subgroups’, Publ. Math. Debrecen 4 (1955–1956), 455–458.CrossRefGoogle Scholar
[5]Huppert, B., Endliche Gruppen (Springer, Berlin, 1967).CrossRefGoogle Scholar
[6]Huppert, B. and Blackburn, N., Finite groups II (Springer, Berlin, 1982).CrossRefGoogle Scholar
[7]Kazarin, L. S., ‘Soluble products of groups’, in: Infinite Groups 94 (eds. de Giovanni, F. and Newell, M.) (de Gruyter, New York, 1995) pp. 111–123.Google Scholar
[8]Kegel, O. H., ‘Produkte nilpotenter Gruppen’, Arch. Math. 12 (1961), 90–93.CrossRefGoogle Scholar
[9]Wielandt, H., ‘Über Produkte von nilpotenten Gruppen’, Illinois J. Math. 2 (1958), 611–618.CrossRefGoogle Scholar
[10]Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).Google Scholar