Semifield Planes Admitting the Quaternion Group Q8

Abstract

We discuss a well-known conjecture that the full automorphism group of a finite projective plane coordinatized by a semifield is solvable. For a semifield plane of order pN (p > 2 is a prime, 4|p − 1) admitting an autotopism subgroup H isomorphic to the quaternion group Q8, we construct a matrix representation of H and a regular set of the plane. All nonisomorphic semifield planes of orders 54 and 134 admitting Q8 in the autotopism group are pointed out. It is proved that a semifield plane of order p4, 4|p−1, does not admit SL(2, 5) in the autotopism group.