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


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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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).

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