Vol. 59, No. 2, 1975

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A characterization of the symplectic groups PSp(2m, q) as rank 3 permutation groups

Arthur Anthony Yanushka

Vol. 59 (1975), No. 2, 611–621

In this paper the following characterization of the symplectic groups PSp(2m,q) for m > 2 as rank 3 permutation groups is obtained:

THEOREM. Let G be a transitive rank 3 group of permutations of a finite set X such that the orbit lengths for Gb, the stabilizer of a point b in X are 1, q(qr2 1)(q 1) and qr1 for integers q > 1 and r > 4. Let b denote the union of b and the Gb orbit of length q(qr2 1)(q 1). Let R(bc) denote ∩{z : b,c z}. Assume R(bc){b,c}, for all distinct pairs of points, b and c. Assume that the pointwise stabilizer of b is transitive on the points unequal to b of R(bc) for cb. Then r is even, q is a prime power and GH, a group of symplectic collineations of projective r 1 space over the finite field of q elements and PSp(r,q)⊲= H.

Mathematical Subject Classification 2000
Primary: 20B10
Received: 21 April 1975
Published: 1 August 1975
Arthur Anthony Yanushka