Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-16T20:39:52.678Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal generating sets for some wreath products of groups

Published online by Cambridge University Press:  17 April 2009

Yeo Kok Chye
Yeo Kok Chye
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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.

Let d(G) denote the minimum of the cardinalities of the generating sets of the group G. Call a generating set of cardinality d(G) a minimal generating set for G. If A is a finitely generated nilpotent group, B a non-trivial finitely generated abelian group and A wr B is their (restricted, standard) wreath product, then it is proved (by explicitly constructing a minimal generating set for A wr B ) that d(AwrB) = max{l+d(A), d(A×B)} where A × B is their direct product.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 1 , August 1973 , pp. 127 - 136
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Curtis, Charles W., Reiner, Irving, Representation theory of finite groups and associative algebras (Pure and Applied Mathematics, XI. Interscience [John Wiley & Sons], New York, London, 1962; reprinted 1966).Google Scholar
[2]Gaschütz, Wolfgang, “Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden”, Math. Z. 60 (1954), 274–286.CrossRefGoogle Scholar
[3]Gaschütz, Wolfgang, “Die Eulersche Funktion endlicher auflösbarer Gruppen”, Illinois J. Math. 3 (1959), 469–476.CrossRefGoogle Scholar
[4]Gaschütz, Wolfgang, “Praefrattinigruppen”, Arch. Math. 13 (1962), 418–426.CrossRefGoogle Scholar
[5]Magnus, Wilhelm, Karrass, Abraham, Solltar, Donald, Combinatorial group theory (Interscience [John Wiley & Sons], New York, London, Sydney, 1966).Google Scholar
[6]McLain, D.H., “Finiteness conditions in locally soluble groups”, J. London Math. Soc. 34 (1959), 101–107.CrossRefGoogle Scholar
[7]Neumann, Hanna, Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar