A projective plane is characterized to a certain extent by the amount of transitivity it possesses. This amounts essentially to saying that the plane is characterized by its group of collineations. The transitive planes that have been most thoroughly studied are the cyclic planes (8; 10; 12; 13; 14). It is believed that all finite cyclic planes are Desarguesian, but it has not been proved that this is the case.

Let Fq = GF(q) denote the finite field of order q = pn, where p is a prime. Consider the group Γ of linear transformations

(1.1) x' = (ax + b)//(cx + d)

with coefficients a, b, c, d ∊ Fq and of determinant 1. The order of Γ is ½q(q2 − 1) or q(q2 − 1) according as q is odd or even, i.e., according as p > 2 or p = 2.

1. Introduction. There are five well-known, two-parameter families of simple finite groups: the unimodular projective group, the symplectic group,1 the unitary group,2 and the first and second orthogonal groups, each group acting on a vector space of a finite number of elements (2; 3).

1, Introduction. The first systematic study of equivalence of indefinite ternary quadratic forms seems to be that of A. Meyer (10) (see also Bachmann (1)). By methods which are often obscure he showed that the number of classes in a genus is a power of 2, the exact power depending on certain quadratic characters associated with the form.