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

Arthur Anthony Yanushka

Vol. 59 (1975), No. 2, 611–621

DOI: 10.2140/pjm.1975.59.611

Abstract

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 G_b, the stabilizer of a point b in X are 1, q(q^r−2 − 1)∕(q − 1) and q^r−1 for integers q > 1 and r > 4. Let b^⊥ denote the union of b and the G_b orbit of length q(q^r−2 − 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 c∉b^⊥. Then r is even, q is a prime power and G≅H, 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

Milestones

Received: 21 April 1975

Published: 1 August 1975