Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-11T06:51:48.379Z Has data issue: false hasContentIssue false
  • English
  • Français

Subnormal subgroups in direct products of groups

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

Published online by Cambridge University Press:  09 April 2009

Peter Hauck
Peter Hauck
Affiliation:
Mathematisches InstitutAlbert-Ludwigs-Universität Albertstraβe 23 b D-7800 Freiburg Federal Republic of, Germany
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 group G is called normally (subnormally) detectable if the only normal (subnormal) subgroups in any direct product G1 × … × Gn of copies of G are just the direct factors Gi. We give an internal characterization of finite subnormally detectable groups and obtain analogous results for associative rings and for Lie algebras. The main part of the paper deals with a study of normally detectable groups, where we verify a conjecture of T. O. Hawkes in a number of special cases.

MSC classification

Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups 20D35: Subnormal subgroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 2 , April 1987 , pp. 147 - 172
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Amayo, R. K. and Stewart, I., Infinite-dimensional Lie algebras (Noordhoff, Leyden, 1974).Google Scholar
[2]Baer, R., ‘Meta ideals’ (Report of a conference on linear algebra (1956), Nat. Acad. Sci. USA, Washington D. C., 1957), pp. 33–52.Google Scholar
[3]Doerk, K. and Hawkes, T. O., Finite soluble groups (in preparation).Google Scholar
[4]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[5]Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachusetts, 1966).Google Scholar
[6]Remak, R., ‘Über die Zerlegung der endlichen Gruppen in direkt unzerlegbare Faktoren’, J. Reine Angew. Math. 139 (1911), 293–308.CrossRefGoogle Scholar
[7]Remak, R., ‘Über die Darstellung der endlichen Gruppen als Untergruppen direkter Produkte’, J. Reine Angew. Math. 163 (1930), 1–44.Google Scholar
[8]Robinson, D. J. S., A course in the theory of groups (Springer, Berlin, 1982).Google Scholar