Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T06:54:50.412Z Has data issue: false hasContentIssue false
  • English
  • Français

On groups in which every subgroup is subnormal of defect at most three

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Gunnar Traustason
Gunnar Traustason
Affiliation:
Christ Church Oxford OX1 1DPEngland e-mail: traustas@ermine.ox.ac.uk
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 this paper we study groups in which every subgroup is subnormal of defect at most 3. Let G be a group which is either torsion-free or of prime exponent different from 7. We show that every subgroup in G is subnormal of defect at most 3 if and only if G is nilpotent of class at most 3. When G is of exponent 7 the situation is different. While every group of exponent 7, in which every subgroup is subnormal of defect at most 3, is nilpotent of class at most 4, there are examples of such groups with class exactly 4. We also investigate the structure of these groups.

MSC classification

Secondary: 20F19: Generalizations of solvable and nilpotent groups 20F45: Engel conditions 20F18: Nilpotent groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 3 , June 1998 , pp. 397 - 420
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bachmuth, S. and Mochizuki, H. Y., ‘Third Engel groups and the Macdonald-Neumann conjecture’, Bull. Austral. Math. Soc. 5 (1971), 379–386.CrossRefGoogle Scholar
[2]Baer, R., ‘Situation der Untergruppen und Struktur der Gruppe’, S. B. Heidelberg Akad. Math. Nat. Klasse 2 (1933), 12–17.Google Scholar
[3]Dedekind, R., ‘Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind’, Math. Ann. 48 (1897), 548–561.CrossRefGoogle Scholar
[4]Golod, E. S., ‘Some problems of Burnside type’, in: Proc. Int. Congr. Math., Moscow, 1966 pp. 284–289, 1968; English translation in Amer. Math. Soc. Transl. Ser. 2, 84 (1969), 83–88.Google Scholar
[5]Gupta, N. D., ‘Third-Engel 2-groups are soluble’, Canad. Math. Bull. 15 (1972), 532–524.CrossRefGoogle Scholar
[6]Gupta, N. D. and Newman, M. F., ‘On metabelian groups’, J. Austral. Math. Soc. 6 (1966), 362–368.CrossRefGoogle Scholar
[7]Gupta, N. D. and Newman, M. F., ‘Third Engel groups’, Bull. Austral. Math. Soc. 40 (1989), 215–230.CrossRefGoogle Scholar
[8]Heineken, H., ‘Engelsche Elemente der Länge drei’, Illinois J. Math. 5 (1961), 681–707.CrossRefGoogle Scholar
[9]Heineken, H., ‘A class of three-Engel groups’, J. Algebra 17 (1971), 341–345.CrossRefGoogle Scholar
[10]Heineken, H. and Mohamed, I. J., ‘A group with trivial centre satisfying the normalizer condition’, J. Algebra 10 (1968), 368–376.CrossRefGoogle Scholar
[11]Kappe, L. C. and Kappe, W. P., ‘On three-Engel groups’, Bull. Austral. Math. Soc. 7 (1972), 391–405.CrossRefGoogle Scholar
[12]Levi, F. W., ‘Groups in which the commutator operation satisfies certain algebraic conditions’, J. Indian Math. Soc. 6 (1942), 87–97.Google Scholar
[13]Mahdavianary, S. K., ‘A special class of three-Engel groups’, Arch. Math. (Basel) 40 (1983), 193–199.CrossRefGoogle Scholar
[14]Roseblade, J. E., ‘On groups in which every subgroup is subnormal’, J. Algebra 2 (1965), 402–412.CrossRefGoogle Scholar
[15]Traustason, G., ‘Engel Lie-algebras’, Quart. J. Math. Oxford Ser. (2) 44 (1993), 355–384.CrossRefGoogle Scholar
[16]Traustason, G., ‘On 4-Engel groups’, J. Algebra 178 (1995), 414–429.CrossRefGoogle Scholar
[17]Vaughan-Lee, M., ‘Engel-4 groups of exponent 5’, Proc. Lond. Math. Soc. (to appear).Google Scholar
[18]Vaughan-Lee, M., The restricted Burnside problem, second edition (Clarendon Press, Oxford, 1993).CrossRefGoogle Scholar
[19]Zorn, M., ‘Nilpotency of finite groups’, Bull. Amer. Math. Soc. 42 (1936), 485–486.Google Scholar