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 Chebyshevskii Sb., 2019, Volume 20, Issue 3, Pages 316–332 (Mi cheb814)  On some $3$-primitive projective planes

O. V. Kravtsova, I. V. Sheveleva

Institute of mathematics and computer sciences of Siberian Federal University (Krasnoyarsk)

Abstract: 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 $i<n$ 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 $n\geq 4$.

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

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00707_a The study was carried out with a grant from the Russian Foundation for Basic Research (project 16-01-00707).

DOI: https://doi.org/10.22405/2226-8383-2018-20-3-316-332  Full text: PDF file (680 kB)

UDC: 519.145
Accepted:12.11.2019

Citation: O. V. Kravtsova, I. V. Sheveleva, “On some $3$-primitive projective planes”, Chebyshevskii Sb., 20:3 (2019), 316–332 Citation in format AMSBIB
\Bibitem{KraShe19} \by O.~V.~Kravtsova, I.~V.~Sheveleva \paper On some $3$-primitive projective planes \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 3 \pages 316--332 \mathnet{http://mi.mathnet.ru/cheb814} \crossref{https://doi.org/10.22405/2226-8383-2018-20-3-316-332} 

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