Semifield planes of rank 2 admitting the group $S_3$

O. V. Kravtsova ; T. V. Moiseenkova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 118-128

Résumé

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.

Détail

Keywords: semifield plane, symmetric group, Baer involution, homology, spread set.
Mots-clés : autotopism group