Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T00:54:53.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Singer Groups

Published online by Cambridge University Press:  20 November 2018

Marshall D. Hestenes
Marshall D. Hestenes*
Affiliation:
Michigan State University, East Lansing, Michigan
Rights & Permissions [Opens in a new window]

Extract

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.

Interest in the Singer groups has arisen in various places. The name itself results from the connection Singer [7] made between these groups and perfect difference sets, and this is closely associated with the geometric property that a Singer group is regular on the points of a projective space. Some information about these groups appears in Huppert's book [3, p. 187]. Singer groups are frequently useful in constructing examples and counterexamples. Our aim in this paper is to make a systematic study of the Singer subgroups of the linear groups, with a particular view to analyzing the examples they provide of Frobenius regular groups. Frobenius regular groups are a class of permutation groups generalizing the Zassenhaus groups, and Keller [5] has shown recently that they provide a new characterization of A6 and M11.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 492 - 513
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Artin, E., The orders of the linear groups, Comm. Pure Appl. Math. 8 (1955), 355–366.Google Scholar
2. Higman, D. G. and McLaughlin, J. E., Geometric ABA-groups, Illinois J. Math. 5 (1961), 382–397.Google Scholar
3. Huppert, B., Endliche Gruppen, Vol. I (Springer-Verlag, New York, 1967).Google Scholar
4. Karzel, H., Bericht iiber projektive Inzidenzgruppen, Jber. Deutsch. Math.-Verein. 67 (1964/65), Abt. 1, 58–92.Google Scholar
5. Keller, G., A characterization of AQ and Mn, J. Algebra 18 (1969), 409–421.Google Scholar
6. Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, N. J., 1964).Google Scholar
7. Singer, J., A theorem infinite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385.Google Scholar