Hostname: page-component-6bf8c574d5-xtvcr Total loading time: 0 Render date: 2025-02-14T11:04:08.919Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite coverings by 2-engel groups

Published online by Cambridge University Press:  17 April 2009

Luise-Charlotte Kappe
Luise-Charlotte Kappe
Affiliation:
Department of Mathematical Sciences, University Center at Binghamton State University of New York, Binghamton, N.Y. 13901, United States of America
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.

Bear's characterisation of central-by-finite groups as groups possessing a finite covering by abelian subgroups if the starting point for this invertigation. We characterise groups with a finite covering by 2-Engel subgroups as groups for which the subgroup of right 2-Engel elements has finite index; and the groups having a finite covering by normal 2-Engel subgroups are exactly the 3-Engel groups among those having a finite covering by 2-Engel subgroups. The second centre of a group having a finite covering by class two subgroups does not necessarily have finite index. However, a group has a finite covering by subgroups in a variety containing all cyclic groups if the margin of thes variety in the group has finite index.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 1 , August 1988 , pp. 141 - 150
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Brodie, M.A., Chamberlain, R.F. and Kappe, L.C., ‘Finite coverings by normal subgroups’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[2]Hall, P., ‘Verbal and Marginal Subgroups’, J. Reine Angew. Math. 182 (1940), 156–157.CrossRefGoogle Scholar
[3]Kappe, L.C. and Kappe, W.P., ‘On three-Engel groups’, Bull. Austral. Math. Soc. 7 (1972), 391–405.CrossRefGoogle Scholar
[4]Kappe, W.P., ‘Die A-Norm einer Gruppe’, Illinois J. Math. 5 (1961), 187–197.CrossRefGoogle Scholar
[5]Macdonald, I.D., ‘Some Examples in the Theory of Groups’, Mathematical Essays Dedicated to A.J. MacIntyre, pp. 263–269 (Ohio University Press, 1970).Google Scholar
[6]Neumann, B.H., ‘Groups covered by permutable subsets’, J. London Math Soc. 29 (1954), 236–248.CrossRefGoogle Scholar
[7]Neumann, B.H., ‘Groups covered by finitely many cosets’, Publ. Math. Debrecen 3 (1954), 227–242.CrossRefGoogle Scholar
[8]Neumann, H., Varieties of groups, Ergeb. Math. Grenzgb. 37 (Springer Verlag, Berlin, 1967).CrossRefGoogle Scholar
[9]Robinson, D.J.S., Finiteness conditions and generalized soluble groups I, Ergeb. Math. Grenzgb. 62 (Springer Verlag, Berlin, 1972).Google Scholar
[10]Schur, J., ‘Über die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen’, J. Reine Angew. Math. 127 (1904), 20–30.Google Scholar
[11]Teague, T.K., ‘On the Engel Margin’, Pacific J. Math. 50 (1974), 205–214.CrossRefGoogle Scholar