Theorem 11.2.3: Let F = F(ta, t, tb) be a monomial semi-field type flock in PG(3,q) with q = pn for some prime p, a = pi, b = pj with 0 < i < j < n. Let d = gcd(i,j,n), and n' = n/d. Then the points DF and lines joining pairs of them are the points and lines of an AG(n',pd).
Pf: If pd > 3, it follows directly from Buekenhout's theorem (Theorem 11.2.2 ) that the points of DF and the lines joining pairs of them form an affine space. In the cases of pd = 2 or 3, we also need to show that parallelism of the lines is an equivalence relation to reach the same conclusion. This was done in Proposition 11.2.4 . Now, by Proposition 11.2.2 , the number of points on each line of this affine space is pd. The total number of points in an affine space AG(n',q') is (q')n'. Since q' = pd and the total number of points is q = pn, we must have the dimension of this affine space be n' = n/d.