Stanley E. Payne (CU-Denver)
Cyclic q-Clans, Herds and the Magic Action

Let \cal F be the set of functions from GF(q) to itself that map zero to zero. C. M. O'Keefe and T. Penttila have constructed an action of P(Gamma)L(2,q) on \cal F which they call the Magic Action. Associated with a q-clan \cal C there is a subset of \cal F called a herd H(\cal C), which is useful if and only if \cal C is not classical. We propose a new definition of "herd" that includes the classical case, and we elucidate the connection between the stabilizer of a herd under the magic action and the collineation group of the associated GQ. We apply these ideas to the "cyclic" q-clans of W. Cherowitzo, C.M. O'Keefe and T. Penttila, including the "new" Adelaide q-clans.