We study the well-known hypothesis of D.R. Hughes that the full collineation group of a finite-order non-Desarguesian semifield projective plane is solvable (see also N.D. Podufalov’s Question 11.76 in the Kourovka Notebook). This hypothesis is reduced to the autotopism group that consists of collineations fixing a triangle. We complete the description of perspectivity-free dihedral and quaternion autotopism subgroups of order 8 in the case of an odd-order semifield plane. A matrix representation and a geometric meaning of generating elements are given together with conditions for the spread set of the plane. Examples of semifield planes of order 81 are presented. The results can be used in the study of semifield planes with autotopism subgroups from J.G. Thompson’s list of minimal simple groups.
Keywords: semifield plane, semifield, spread set, homology, autotopism group, Baer involution, quaternion group, dihedral group, Hughes problem
Received October 1, 2024
Revised October 8, 2024
Accepted October 14, 2024
Funding Agency: This work was supported by the Krasnoyarsk Mathematical Center, which is financed by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2024-1429).
Olga Vadimovna Kravtsova, Dr. Phys.-Math. Sci., Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: ol71@bk.ru
Daria Sergeevna Skok, Institute of Computational Modeling SB RAS, Krasnoyarsk, 660041 Russia, e-mail: skokdarya@yandex.ru
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Cite this article as: O.V. Kravtsova, D.S. Skok. Non-Abelian autotopism subgroups of order 8 of semifield projective planes. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 90–100.
