Annali di Matematica Pura ed Applicata

, Volume 158, Issue 1, pp 33–50 | Cite as

Spreads and classes of maximal subgroups ofGL n (q),SL n (q),PGL n (q) andPSL n (q)

  • R. H. Dye


If r divides n then the points of PG(n−1, q) can be partitioned by the (r−1)-subspaees of a classical spread Sr. The underlying finite geometry of this configuration, in particular the orbits of lines, is used to prove that if r is a proper prime divisor of n then the stabilizers of Sr in PGLn(g) and PSLn(q) are maximal subgroups of PGLn(q) and PSLn(q respectively. Special attention is needed for the case of PSLn(q) when n/r=2 and r divides q− 1. An explicit description is found for the stablizers.


Maximal Subgroup Prime Divisor Explicit Description Finite Geometry Classical Spread 


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  1. [1]
    M. Aschbacher,On the maximal subgroups of the finite classical groups, Invent. Math.,76 (1984), pp. 469–514.Google Scholar
  2. [2]
    M. Aschbacher,Subgroup structure of finite groups, Proc. of Rutgers Group Theory year, 1983–1984, CUP, Cambridge, 1984.Google Scholar
  3. [3]
    P. J. Cameron -W. M. Kantor, 2-transitive and antiflag transitive collineation groups of finite projective spaces, J. Algebra,60 (1979), pp. 384–422.Google Scholar
  4. [4]
    J. Dieudonné,La géométrie des groupes classiques, 3rd edition, Springer-Verlag, Berlin/ Heidelberg/New York, 1971.Google Scholar
  5. [5]
    R. H. Dye,Partitions and their stabilizers for line complexes and quadrics, Ann. Mat. Pura Appl., (V),114 (1977), pp. 173–194.Google Scholar
  6. [6]
    R. H. Dye,A maximal subgroup of PSp 6 (2m)related to a spread, J. Algebra,84 (1983), pp. 128–135.Google Scholar
  7. [7]
    R. H. Dye,Maximal subgroup of symplectic groups stabilizing spreads, J. Algebra,87 (1984), pp. 493–509.Google Scholar
  8. [8]
    R. H. Dye,Maximal subgroups of PSp 6n(q) stabilising spreads of totally isotropic planes, J. Algebra,99 (1986), pp. 191–209.Google Scholar
  9. [9]
    R. H. Dye,Maximal subgroups of finite orthogonal groups stabilizing spreads of lines, J. London Math. Soc., (2),33 (1986), pp. 279–293.Google Scholar
  10. [10]
    N. Jacobson,Lectures in abstract algebra, Vol. III:Theory of fields and Galois Theory, D. Van Nostrand Co. Inc., Princeton, 1964.Google Scholar
  11. [11]
    M. W. Liebeck,On the orders of maximal subgroups of the finite classical group, Proc. London Math. Soc.,50 (1985), pp. 426–446.Google Scholar
  12. [12]
    LiShangzhi,Maximal subgroups in classical groups over arbitrary fields, Proc. of Symposia in Pure Mathematics (to appear).Google Scholar
  13. [13]
    LiShangzhi,Overgroups in GL(nr, F) of certain subgroups of SL(n, K) (manuscript).Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1991

Authors and Affiliations

  • R. H. Dye
    • 1
  1. 1.School of Mathematics The UniversityEngland

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