Definitions: Flock, Linear Flock, Star Flock, Herd, Carrier


Proposition 9.2: The flocks of all thick cones in PG(3,4) that have flocks are linear. The flocks of any cone with more than two points in its carrier are star flocks.

Pf: There are 4 planes in a flock in PG(3,4), any two of which must meet in a common line. Since for q = 4, q -[q+2/2] = 1, all flocks of thick cones are linear by Theorem 9.1 [backbutton].

There are 6 permutations of GF(4) which fix 0, the 3 scalar multiples of f(t) = t and the 3 scalar multiples of f(t) = t2. Thus, if there are more than two points in the carrier of a cone, the herd of any flock of the cone would have to contain two permutations which are scalar multiples of each other. By Theorem 8.1.1 [backbutton] the flock must be a star flock.


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