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Maximal subgroups of symmetric groups defined on projective spaces over finite fields

Abstract

Let PΓL(n, q) be a complete projective group of semilinear transformations of the projective space P(n−1, q) of projective degree n−l over a finite field of q elements; we consider the group in its natural 2-transitive representation as a subgroup of the symmetric group S(P*(n−1, q)) on the setp*(n−1),q=p(n−1,q)/{O}. In the present note we show that for arbitrary n satisfying the inequality n>4[(qn−1)/(qn−1−1)] [in particular, for n>4(q +l)] and for an arbitrary substitutiong ε s (p*(n−1,q))∖pΓL(n,q) the group 〈PΓL(n,q), g〉 contains the alternating group A(P* (n−1,q)). Forq=2, 3 this result is extended to all n≥3.

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Literature cited

  1. 1.

    E. Artin, Geometric Algebra, Wiley, New York (1957).

  2. 2.

    D. H. Lehmer, Tables of Prime Numbers from 1 to 10,006,721, Carnegie Institution of Washington, Publication No. 165, Washington (1914).

  3. 3.

    M. Hall, The Theory of Groups, Macmillan, New York (1959).

  4. 4.

    C. F. Luther, “Concerning primitive groups of class U, II,” Amer. J. Math.,55, No. 4, 611–618 (1933).

  5. 5.

    W. A. Manning, “The degree and class of multiple-transitive groups, III,” Trans. Amer. Math. Soc.,35, 585–589 (1933).

  6. 6.

    H. Wielandt, Finite Permutation Groups, Academic Press, New York (1964).

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Additional information

Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 91–100, July, 1974.

The author expresses his sincere thanks to M. M. Glukhov for his interest in his work.

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Pogorelov, B.A. Maximal subgroups of symmetric groups defined on projective spaces over finite fields. Mathematical Notes of the Academy of Sciences of the USSR 16, 640–645 (1974). https://doi.org/10.1007/BF01098818

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Keywords

  • Projective Space
  • Symmetric Group
  • Maximal Subgroup
  • Finite Field
  • Projective Group