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Characterization of classical groups
by orbit sizes on the natural module


Author: Martin W. Liebeck
Journal: Proc. Amer. Math. Soc. 124 (1996), 2961-2966
MSC (1991): Primary 20G40
DOI: https://doi.org/10.1090/S0002-9939-96-03505-8
MathSciNet review: 1343709
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Abstract: We show that if $V$ is a finite vector space, and $G$ is a subgroup of $P\Gamma L(V)$ having the same orbit sizes on 1-spaces as an orthogonal or unitary group on $V$, then, with a few exceptions, $G$ is itself an orthogonal or unitary group on $V$.


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  • [Ab] S. Abhyankar, ``Again nice equations for nice groups'', Proc. Amer. Math. Soc. 124 (1996), 3567--3576.
  • [As] M. Aschbacher, ``On the maximal subgroups of the finite classical groups'', Invent. Math. 76 (1984), 469-514. MR 86a:20054
  • [At] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of finite groups, Oxford University Press, 1985. MR 88g:20025
  • [FK] D.A. Foulser and M.J. Kallaher, ``Solvable, flag-transitive, rank 3 collineation groups'', Geom. Ded. 7 (1978), 111-130. MR 57:12263
  • [GPPS] R. Guralnick, T. Penttila, C.E. Praeger and J. Saxl, ``Linear groups with orders having certain primitive prime divisors'', preprint, University of Western Australia, 1994.
  • [He] C. Hering, ``Transitive linear groups and linear groups which contain irreducible subgroups of prime order'', Geom. Ded. 2 (1974), 425-460. MR 49:439
  • [At2] C. Jansen, K. Lux, R.A. Parker and R.A. Wilson, Atlas of modular characters, Oxford University Press, 1995.
  • [JP] W. Jones and B. Parshall, ``On the 1--cohomology of finite groups of Lie type'', Proc. Conf. on Finite Groups, eds. W. Scott and F. Gross, Academic Press, (1976). MR 53:8272
  • [Kl] P.B. Kleidman, The subgroup structure of some finite simple groups, Ph.D. Thesis, University of Cambridge, 1987.
  • [KL] P.B. Kleidman and M.W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Series 129, Cambridge University Press, Cambridge, 1990. MR 91g:20001
  • [Li] M.W. Liebeck, ``The affine permutation groups of rank three'', Proc. London Math. Soc. 54 (1987), 477-516. MR 88m:20004
  • [LPS] M.W. Liebeck, C.E. Praeger and J. Saxl, ``The maximal factorizations of the finite simple groups and their automorphism groups'', Mem. Amer. Math. Soc. 86, No. 432 (1990), 1-151. MR 90k:20048
  • [Zs] K. Zsigmondy, ``Zur Theorie der Potenzreste'', Monatsh. fur Math. und Phys. 3 (1892), 265--284.

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Additional Information

Martin W. Liebeck
Affiliation: Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom
Email: m.liebeck@ic.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-96-03505-8
Received by editor(s): March 20, 1995
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1996 American Mathematical Society

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