**Theorem 11.2.3**:* Let F = F(t ^{a}, t, t^{b}) be a monomial semi-field type flock in PG(3,q) with q = p^{n} for some prime p, a = p^{i}, b = p^{j} with 0 < i < j < n. Let d = gcd(i,j,n), and n' = n/d. Then the points D_{F} and lines joining pairs of them are the points and lines of an AG(n',p^{d}).*

*Pf*: If p^{d} > 3, it follows directly from Buekenhout's theorem (Theorem 11.2.2 ) that the points of D_{F} and the lines joining pairs of them form an affine space. In the cases of p^{d} = 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 p^{d}. The total number of points in an affine space AG(n',q') is (q')^{n'}. Since q' = p^{d} and the total number of points is q = p^{n}, we must have the dimension of this affine space be n' = n/d.