On some $3$-primitive projective planes

O. V. Kravtsova ; I. V. Sheveleva

Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 316-332

Résumé

We evolve an approach to construction and classification of semifield projective planes with the use of the linear space and spread set. This approach is applied to the problem of existance for a projective plane with the fixed restrictions on collineation group. A projective plane is said to be semifield plane if its coordinatizing set is a semifield, or division ring. It is an algebraic structure with two binary operation which satisfies all the axioms for a skewfield except (possibly) associativity of multiplication. A collineation of a projective plane of order $p^{2n}$ ($p>2$ be prime) is called Baer collineation if it fixes a subplane of order $p^n$ pointwise. If the order of a Baer collineation divides $p^n-1$ but does not divide $p^i-1$ for $i$ then such a collineation is called $p$-primitive. A semifield plane that admit such collineation is a $p$-primitive plane. M. Cordero in 1997 construct 4 examples of $3$-primitive semifield planes of order $81$ with the nucleus of order $9$, using a spread set formed by $2\times 2$-matrices. In the paper we consider the general case of $3$-primitive semifield plane of order $81$ with the nucleus of order $\leq 9$ and a spread set in the ring of $4\times 4$-matrices. We use the earlier theoretical results obtained independently to construct the matrix representation of the spread set and autotopism group. We determine $8$ isomorphism classes of $3$-primitive semifield planes of order $81$ including M. Cordero examples. We obtain the algorithm to optimize the identification of pair-isomorphic semifield planes, and computer realization of this algorithm. It is proved that full collineation group of any semifield plane of order $81$ is solvable, the orders of all autotopisms are calculated. We describe the structure of $8$ non-isotopic semifields of order $81$ that coordinatize $8$ non-isomorphic $3$-primitive semifield planes of order $81$. The spectra of its multiplicative loops of non-zero elements are calculated, the left-, right-ordered spectra, the maximal subfields and automorphisms are found. The results obtained illustrate G. Wene hypothesis on left or right primitivity for any finite semifield and demonstrate some anomalous properties. The methods and algorithsm demonstrated can be used for construction and investigation of semifield planes of odd order $p^n$ for $p\geq 3$ and $n\geq 4$.

Détail

Keywords: semifield plane, collineation, autotopism, Baer subplane.