Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-21T07:14:08.025Z Has data issue: false hasContentIssue false
  • English
  • Français

On an infinite integral linear group

Published online by Cambridge University Press:  17 April 2009

I.H. Farouqi  and
Hanna Neumann
I.H. Farouqi
Affiliation:
Department of Mathematics, The University of Karachi, Karachi, Pakistan.
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.

This paper investigates the normal subgroup structure of the automorphism group Γ of a free abelian group A of countably infinite rank. The finitary automorphisms, that is those acting non-trivially only on a direct summand of A of finite rank, form a normal subgroup Φ of Γ; the sublattice of all normal subgroups of Γ contained in Φ is in fact the sublattice of normal subgroups of Φ and has a quite transparent structure. By contrast there is a profusion of normal subgroups of Γ not contained in Φ. For example, the collection of certain types of these normal subgroups, defined as generalizations of the congruence subgroups of finite dimensional integral linear groups, if partially ordered by inclusion, can be shown to contain infinitely many chains of the order type of the continuum.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 4 , Issue 3 , June 1971 , pp. 321 - 342
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Brenner, J.L., “The linear homogeneous group, III”, Ann. of Math. 71 (1960), 210–223.CrossRefGoogle Scholar
[2]Fuchs, L., Abelian groups (Pergamon Press, Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 3rd ed. 1960).Google Scholar
[3]Kaplansky, Irving, Infinite abelian groups (The University of Michigan Press, Ann Arbor, 1954).Google Scholar
[4]Maxwell, George, “Infinite general linear groups over rings”, Trans. Amer. Math. Soc. 151 (1970), 371–375.Google Scholar
[5]Mennicke, Jens L., “Finite factor groups of the unimodular group”, Ann. of Math. 81 (1965), 31–37.Google Scholar
[6]Rosenberg, Alex, “The structure of the infinite general linear group”, Ann. of Math. 68 (1958), 278–294.CrossRefGoogle Scholar
[7]Schreier, J. und Ulam, S., “Über die Permutationsgruppe der natürlichen Zahlenfolge”, Studia Math. 4 (1933), 134–141.CrossRefGoogle Scholar