A CHARACTERIZATION OF THE SYMPLECTIC GROUPS PSp(2tn,q) AS RANK 3 PERMUTATION GROUPS

In this paper the following characterization of the sym-plectic groups PSp(2m,q) for m> 2 as rank 3 permutation groups is obtained: THEOREM. Let G be a transitive rank 3 group of per-mutations of a finite set X such that the orbit lengths for G ύ9 the stabilizer of a point 6 in X are 1, q{qr~z- l)l(q- 1) and q*"1 for integers q> 1 and r> 4. Let b1 denote the union of b and the Gh orbit of length q(qr~2 — l)/(g —1) Let R(bc) denote Π{z±:b>cez1}. Assume R(bc) Φ {6, c}, for all distinct pairs of points, b and c. Assume that the pointwise stabilizer of b1 is transitive on the points unequal to b of R{bc) for c £ b1. Then r is even, q is a prime power and G £* H, a group of symplectic collίneations of projective r — 1 space over the finite field of q elements and PSp(r,q)<3H