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(q^{2m−2} − 1)∕(q − 1) and
q^{2m−1} 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(q^{r−2} − 1)∕(q − 1) and q^{r−1} for
integers q > 1 and r > 4. Let x^{⊥} denote the union of the orbits of length 1 and
q(q^{r−2} − 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 x^{⊥}∩y^{⊥} for y∉x^{⊥} 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).
