Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T13:31:46.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups of odd order in which every subnormal subgroup has defect at most two

Part of: Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

John Cossey
John Cossey
Affiliation:
Department of Mathematics A.N.U. GPO Box 4 Canberra, ACT 2601, Australia
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.

In 1980, McCaughan and Stonehewer showed that a finite soluble group in which every subnormal subgroup has defect at most two has derived length at most nine and Fitting length at most five, and gave an example of derived length five and Fitting length four. In 1984 Casolo showed that derived length five and Fitting length four are best possible bounds.

In this paper we show that for groups of odd order the bounds can be improved. A group of odd order with every subnormal subgroup of defect at most two has derived and Fitting length at most three, and these bounds are best possible.

MSC classification

Secondary: 20D35: Subnormal subgroups 20E34: General structure theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 2 , October 1991 , pp. 331 - 342
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Curtis, C. W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, (Pure and Applied Mathematics 11, Interscience Publishers, New York, London, 1962).Google Scholar
[2]Casolo, C., ‘Gruppi finite risolubili in cui tutti i sottogrupi subnormali hanno difetto al pui 2’ Rend. Sem. Mat. Univ. Padova 71 (1984), 257–271.Google Scholar
[3]Gaschütz, W., ‘Grupprn in denen das Normalteilersein transitiv ist’, J. Reine Angew. Math., 198 (1957), 87–92.CrossRefGoogle Scholar
[4]Gaschütz, W., ‘Endliche Gruppen mit treuen absolut-irreduziblen Darstellungen’, Math. Nachr. 12 (1954), 253–255.CrossRefGoogle Scholar
[5]Heineken, H., ‘A class of three Engel groups’, J. Algebra 17 (1971), 341–345.CrossRefGoogle Scholar
[6]Higman, G., ‘Complementation of abelian normal subgroups’, Publ. Math. Debrecen 4 (1955–1956), 455–458.CrossRefGoogle Scholar
[7]Huppert, B., Endliche Gruppen I, (Die Grundlehren der Mathematischen Wissenschaften, Bd 134, Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[8]Huppert, B. and Blackburn, N., Finite Groups II, (Die Grundlehren der Mathematischen Wissenschaften, Bd 242, Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[9]Mahdavianary, S. K., ‘a special class of three Engel groups’, Arch. Math. 40 (1983), 193–199.CrossRefGoogle Scholar
[10]McCaughan, D. J. and Stonehewer, S., ‘Finite soluble groups whose subnormal subgroups have defect at most two’, Arch. Math. 35 (1980), 56–60.CrossRefGoogle Scholar
[11]Peng, T. A., ‘Finite groups with pronormal subgroups’, Amer. Math. Soc. 20 (1969), 232–234.CrossRefGoogle Scholar
[12]Robinson, D. J. S., ‘A note on finite groups in which normality is transitive’, Proc. Amer. Math. Soc. 19 (1968), 933–937.CrossRefGoogle Scholar
[13]Robinson, D. J. S., A Course in the Theory of Groups, (Graduate Texts in Mathematics 80, Springer-Verlag, New York, Heidelberg, Berlin, 1982).CrossRefGoogle Scholar
[14]Taunt, D. R., ‘On A-groups’, Proc. Cambridge Phil. Soc. 45 (1949), 24–42.CrossRefGoogle Scholar
[15]Zacher, G., ‘Caratterizzazione dei t-gruppi finiti risolubili’, Ricerche Mat. 1 (1952), 287–294.Google Scholar