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.