**Proposition 11.2.2**: *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). Then through every line of intersection of two planes of F there pass exactly p^{d} planes of F. Dually, all lines containing two points of D_{F} contain exactly p^{d} points of D_{F}.*

*Pf*: We shall work in the dual setting. To prove this proposition, we will exhibit a group of collineations of PG(3,q) which stabilize the plane W'=0 and the set D_{F} which acts doubly transitively on D_{F}. Thus, any two distinct points of D_{F} can be mapped to any other pair of distinct points, so any line containing two points of D_{F} can be mapped to any other line containing two points of D_{F}. Therefore, every line containing two points of D_{F} must contain the same number of D_{F} points, which by Proposition 11.2.1 is p^{d}.

Consider the group **G** of collineations generated by the matrices:

where , in GF(q), 0. Clearly, the collineations of **G** stabilize W' = 0. Since (t^{a},t,t^{b},1)M(,) = ((t+)^{a},(t+),(t+)^{b},1), we see that the set D_{F} is stabilized by **G**. Finally, to show that **G** acts doubly transitively on D_{F}, we need only show that two specific points can be mapped to any two distinct points of D_{F}. The collineation which maps (0,0,0,1) to (t^{a},t,t^{b},1) and (1,1,1,1) to (s^{a},s,s^{b},1) with ts is given by M(s-t,t).