Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-04T21:05:24.091Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal subgroups of infinite dimensional general linear groups

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

Dugald Macpherson
Dugald Macpherson
Affiliation:
School of Mathematical Sciences Queen Mary and Westfield CollegeMile End RoadLondon E1 4NS, England
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 k be an infinite cardinal, F a field, and let GL(k, F) be the group of all non-singular linear transformations on a ki-dimensional vector space V over F. Various examples are given of maximal subgroups of GL(k, F). These include (i) stabilizers of families of subspaces of V which are like filters or ideals on a set, (ii) almost stabilizers of certain subspaces of V, (iii) almost stabilizers of a direct decomposition of V into two k-dimensional subspaces.

It is also noted that GL(k, F) is not the union of any chain of length k of proper subgroups.

MSC classification

Secondary: 20B07: General theory for infinite groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 3 , December 1992 , pp. 338 - 351
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Brazil, M., Covington, J. A., Penttila, T., Praeger, C. E., Woods, A., ‘Maximal subgroups of infinite symmetric groups’, preprint.Google Scholar
[2]Dieudonné, J., ‘Les determinants sur un corps non commutatif’, Bull. Soc. Math. France 71 (1943), 27–45.CrossRefGoogle Scholar
[3]Evans, D. M., ‘Subgroups of small index infinite general linear groups’, Bull. London Math. Soc. 18 (1986), 587–590.CrossRefGoogle Scholar
[4]Jacobson, N., Lectures in abstract algebra, II Linear algebra (Van Nostrand, Toronto, 1953).Google Scholar
[5]Jech, T., Set theory (Academic Press, New York, 1978).Google Scholar
[6]Johnson, R. E., ‘Equivalence rings’, Duke Math. J. 15 (1948), 787–793.CrossRefGoogle Scholar
[7]Macpherson, H. D., ‘Large subgroups of infinite symmetric groups’, preprint.Google Scholar
[8]Macpherson, H. D., Neumann, P. M., ‘Subgroups of infinite symmetric groups’, J. London Math. Soc. (2) 42 (1990), 64–84.CrossRefGoogle Scholar
[9]Richmann, F., ‘Maximal subgroups of infinite symmetric groups’, Canad. Math. Bull. 10 (1967), 375–381.CrossRefGoogle Scholar
[10]Rosenberg, A., ‘The structure of the infinite general linear group’, Ann. of Math. 68 (1958), 278–294.CrossRefGoogle Scholar
[11]Semmes, S. W., ‘Infinite symmetric groups, maximal subgroups and filters’, Abstracts Amer. Math. Soc. 3 (1982), 38.Google Scholar
[12]Zelinsky, D., ‘Every linear transformation is a sum of nonsingular ones’, Proc. Amer. Math. Soc. 5 (1954), 627–630.CrossRefGoogle Scholar