Skip to main content Accessibility help
×
Home

On the faithful representations, of degree 2n, of certain extensions of 2-groups by orthogonal and symplectic groups

  • S. P. Glasby (a1)

Abstract

If R is a 2-group of symplectic type with exponent 4, then R is isomorphic to the extraspecial group , or to the central product 4 o 21+2n of a cyclic group of order 4 and an extraspecial group, with central subgroups of order 2 amalgamated. This paper gives an explicit description of a projective representation of the group A of automorphisms of R centralizing Z(R), obtained from a faithful representation of R of degree 2n. The 2-cocycle associated with this projective representation takes values which are powers of −1 if R is isomorphic to and powers of otherwise. This explicit description of a projective representation is useful for computing character values or computing with central extensions of A. Such central extensions arise naturally in Aschbacher's classification of the subgroups of classical groups.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the faithful representations, of degree 2n, of certain extensions of 2-groups by orthogonal and symplectic groups
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the faithful representations, of degree 2n, of certain extensions of 2-groups by orthogonal and symplectic groups
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the faithful representations, of degree 2n, of certain extensions of 2-groups by orthogonal and symplectic groups
      Available formats
      ×

Copyright

References

Hide All
[1]Aschbacher, M., ‘On the maximal subgroups of the finite classical groups’, Invent. Math. 76 (1984), 469514.
[2]Dieudonné, J., Sur les groupes classique (Hermann, Paris, 1984).
[3]Gérardin, P., ‘Weil representations associated to finite fields’, J. Algebra 46 (1977), 54101.
[4]Glasby, S. P. and Howlett, R. B., ‘Extraspecial towers and Weil representations’, J. Algebra 151 (1992), 236260.
[5]Gorenstein, D., Finite groups (Chelsea, New York, 1980).
[6]Greiss, R. L., ‘Automorphisms of extra special groups and nonvanishing degree 2 cohomology’, Pacific J. Math. 48 (1973), 402422.
[7]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).
[8]Kleidman, P. and Liebeck, M., The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series 129 (Cambridge Univ. Press, Cambridge, 1990).
[9]Schur, I., ‘Über die Darstellung der endlichen Gruppen durch gebrochene linear Substitutionen’, J. Reine Angew. Math. 132 (1907), 85137.
[10]Suzuki, M., Group Theory II (Springer, New York, 1986).
[11]Taylor, D. E., The geometry of the classical groups, (Helderman, Berlin, 1992).
[12]Ward, H. N., ‘Representations of symplectic groups’, J. Algebra 20 (1974), 182195.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

On the faithful representations, of degree 2n, of certain extensions of 2-groups by orthogonal and symplectic groups

  • S. P. Glasby (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed