O.V. Kravtsova, T.V. Moiseenkova. Semifield planes of rank 2 admitting the group  $S_3$ ... P. 118-128

One of the classical problems in projective geometry is to construct an object from known constraints on its automorphisms. We consider finite projective planes coordinatized by a semifield, i.e., by an algebraic system satisfying all axioms of a skew-field except for the associativity of multiplication. Such a plane is a translation plane admitting a transitive elation group with an affine axis. Let $\pi$ be a semifield plane of order $p^{2n}$ with a kernel containing $GF(p^n)$ for prime $p$, and let the linear autotopism group of $\pi$ contain a subgroup $H$ isomorphic to the symmetric group $S_3$. For the construction and analysis of such planes, we use a linear space and a spread set, which is a special family of linear mappings. We find a matrix representation for the subgroup $H$ and for the spread set of a semifield plane if $p=2$ and if $p>2$. We also study the existence of central collineations in $H$. It is proved that a semifield plane of order $3^{2n}$ with kernel $GF(3^n)$ admits no subgroups isomorphic to $S_3$ in the linear autotopism group. Examples of semifield planes of order 16 and 625 admitting $S_3$ are found. The obtained results can be generalized for semifield planes of rank greater than 2 and can be applied, in particular, for studying the known hypothesis that the full collineation group of any finite non-Desarguesian semifield plane is solvable.

Keywords: semifield plane, autotopism group, symmetric group, Baer involution, homology, spread set

Received July 25, 2019

Revised October 7, 2019

Accepted October 14, 2019

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00566 А).

Olga Vadimovna Kravtsova, Cand. Sci. (Phys.-Math.), Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: ol71@bk.ru

Tatyana Vladimirovna Moiseenkova, Cand. Sci. (Phys.-Math.), Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: tanya-mois11@yandex.ru

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Cite this article as: O.V.Kravtsova, T.V.Moiseenkova. Semifield planes of rank 2 admitting the group $S_3$, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 118–128.