Vol. 72, No. 1, 1977

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A characterization of PSp(2m,q) and PΩ(2m + 1,q) as rank 3 permutation groups

Arthur Anthony Yanushka

Vol. 72 (1977), No. 1, 273–281

This paper characterizes the projective symplectic groups PSp(2m,q) and the projective orthogonal groups PΩ(2m + 1,q) as the only transitive rank 3 permutation groups G of a set X for which the pointwise stabilizer of G has orbit lengths 1, q(q2m2 1)(q 1) and q2m1 under a relatively weak hypothesis about the pointwise stabilizer of a certain subset of X. A precise statement is

Theorem. Let G be a transitive rank 3 group of permutations of a set X such that the orbit lengths for the pointwise stabilizer are 1, q(qr2 1)(q 1) and qr1 for integers q > 1 and r > 4. Let x denote the union of the orbits of length 1 and q(qr2 1)(q 1). Let R(xy) denote ∩{z : x,y z}. Assume R(xy){x,y} for y x−{x}. Assume that the pointwise stabilizer of xy for yx does not fix R(xy) pointwise. Then r is even, q is a prime power and G is isomorphic to either a group of symplectic collineations of projective (r 1) space over GF(q) containing PSp(r,q) or a group of orthogonal collineations of projective r space over GF(q) containing PΩ(r + 1,q).

Mathematical Subject Classification 2000
Primary: 20B10
Received: 17 July 1975
Revised: 9 July 1976
Published: 1 September 1977
Arthur Anthony Yanushka