Summary
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.
Article PDF
Similar content being viewed by others
References
M. Aschbacher,On the maximal subgroups of the finite classical groups, Invent. Math.,76 (1984), pp. 469–514.
M. Aschbacher,Subgroup structure of finite groups, Proc. of Rutgers Group Theory year, 1983–1984, CUP, Cambridge, 1984.
P. J. Cameron -W. M. Kantor, 2-transitive and antiflag transitive collineation groups of finite projective spaces, J. Algebra,60 (1979), pp. 384–422.
J. Dieudonné,La géométrie des groupes classiques, 3rd edition, Springer-Verlag, Berlin/ Heidelberg/New York, 1971.
R. H. Dye,Partitions and their stabilizers for line complexes and quadrics, Ann. Mat. Pura Appl., (V),114 (1977), pp. 173–194.
R. H. Dye,A maximal subgroup of PSp 6 (2m)related to a spread, J. Algebra,84 (1983), pp. 128–135.
R. H. Dye,Maximal subgroup of symplectic groups stabilizing spreads, J. Algebra,87 (1984), pp. 493–509.
R. H. Dye,Maximal subgroups of PSp 6n (q) stabilising spreads of totally isotropic planes, J. Algebra,99 (1986), pp. 191–209.
R. H. Dye,Maximal subgroups of finite orthogonal groups stabilizing spreads of lines, J. London Math. Soc., (2),33 (1986), pp. 279–293.
N. Jacobson,Lectures in abstract algebra, Vol. III:Theory of fields and Galois Theory, D. Van Nostrand Co. Inc., Princeton, 1964.
M. W. Liebeck,On the orders of maximal subgroups of the finite classical group, Proc. London Math. Soc.,50 (1985), pp. 426–446.
LiShangzhi,Maximal subgroups in classical groups over arbitrary fields, Proc. of Symposia in Pure Mathematics (to appear).
LiShangzhi,Overgroups in GL(nr, F) of certain subgroups of SL(n, K) (manuscript).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dye, R.H. Spreads and classes of maximal subgroups ofGL n (q),SL n (q),PGL n (q) andPSL n (q) . Annali di Matematica pura ed applicata 158, 33–50 (1991). https://doi.org/10.1007/BF01759298
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01759298