Dihedral and Quaternion Autotopism Subgroups of Semifield Projective Planes of Order 𝑝4

In 1959, D.R. Hughes conjecture that the full collineation group of any finite non-Desarguesian semifield projective plane is solvable (see also the question 11.76 by N.D. Podufalov in Kourovka notebook). The spread set method is useful to exclude some simple non-Abelian groups from the list of possible autotopism subgroups (collineations fixing a triangle) or for constructing the examples of semifield planes with certain autotopism subgroup. The present paper continues the series of results on 2-subgroups and 2-elements in an autotopism group. The natural restrictions from previous papers allow us to complete the description of dihedral and quaternion autotopism subgroup of order 8, together with their geometrical sense. For a semifield projective plane of odd order and 4-dimensional over the center, the matrix representation of the spread set is determined, depending on the characteristic of prime field. It is proven that the dihedral autotopism group of order 8 necessarily contains the perspectivities and therefore cannot be a subgroup of any simple non-Abelian group. For the case of a quaternion subgroup without perspectivities, examples of semifield projective planes of order 81 and 2401 are constructed, up to isomorphism. The list of exceptions complements the classical results of H. L¨uneburg etc. on projective special linear collineation groups. The method used and the described algorithms allow for generalization to the case of a different dimension or a different order.